Azumaya Semigroup Algebras

Abstract

A semigroup is called almost idempotent-free if it has exactly one nonzero ${\mathcal {J}}$-class containing idempotents. It is mainly investigated when almost idempotent-free semigroup algebras are Azumaya algebras. Necessary and sufficient conditions for semigroup algebras of an almost idempotent-free semigroup with certain properties to be Azumaya are obtained.

1 Introduction

Recall that an algebra ${\mathfrak {A}}$ with unity over a commutative ring R is called separable if ${\mathfrak {A}}$ is a projective left ${\mathfrak {A}}\otimes _R{\mathfrak {A}}^o$-module under the action given by $(\sum _i x_i\otimes y_i)a=\sum _ix_iay_i$ for all $a,x_i\in {\mathfrak {A}}, y_i\in {\mathfrak {A}}^o$, where ${\mathfrak {A}}^o$ is the opposite algebra of ${\mathfrak {A}}; {\mathfrak {A}}$ is called an Azumaya algebra if it is separable over its center $Z({\mathfrak {A}})$. For separable algebra, the reader can be referred to the references [5, 16]. If R is a commutative ring with unity $1_R$ and G a finite group of order n, then the group ring R[G] is separable if and only if $n1_R$ is invertible in R. This is a well-known generalization of Maschke’s theorem. It is worthy to point out that if $n1_R$ is invertible in R, then R[G] is an Azumaya algebra. So, R[G] is separable if and only if R[G] is Azumaya. On the other hand, Azumaya algebras form a class of very well-behaved PI-algebras that allow development of the Brauer group classification of commutative rings (see [6, 17]). They also prove to be very useful as an efficient tool in the study of general algebras satisfying polynomial identities. We refer to [2] for an important application of this type.

Z-separability of the integral semigroup ring Z[S] of an arbitrary finite semigroup S was first studied by Shapiro in [19]. Then Cheng [3] obtained a description of R-separable ring R[S] for an arbitrary commutative coefficient ring R. DeMeyer and Hardy independently considered this problem in [7]. An extension of these results to the class of so-called excellent extensions was given by Okniński in [12]. In [13], Okniński and Van Oystaeyen gave some necessary and sufficient conditions for a cancellative monoid algebra to be an Azumaya algebra. In a sequence of papers, Van Oystaeyen studied group-graded Azumaya algebras (see, [14, 15]).

A semigroup S is called idempotent-free if S has no idempotents except possibly identity and zero elements; S is called almost idempotent-free if it has exactly one nonzero ${\mathcal {J}}$-class containing idempotents. Okniński pointed out that for a finitely generated semigroup S and a field K, K[S] is an Azumaya algebra if and only if $K[S]\cong \oplus _{i=1}^nK_0[S_i]$ and each $K_0[S_i]$ is an Azumaya algebra, where $S_i$ are almost idempotent-free semigroups (see [11, Corollary 17,  p. 325]). So, it is important to probe when semigroup algebras of almost idempotent-free semigroups are Azumaya algebras. Because of this view, Okniński raised the problem ([11, Problem 28, p. 332]):

Problem 1.1

Characterize Azumaya algebras K[S] of almost idempotent-free semigroups.

He conjectured that the equivalences of [11, Proposition 8,  p. 318] can be extended to idempotent-free semigroups; that is,

Conjecture 1.2

Let S be an idempotent-free monoid with the group of units U. Then the following statements are equivalent:

• (1) K[S] is an Azumaya algebra;

• (2) $K[S]=Z(K[S])K[U]$ and K[U] is an Azumaya algebra;

• (3) $C_{K[S]}(U)$ (the centralizer of U) $=Z(K[S])$ and K[U] is an Azumaya algebra.

Furthermore, in his monograph, he raised [11, Problem 29, p. 332]:

Problem 1.3

Assume that S is a finite semigroup. Is an Azumaya algebra K[S] always isomorphic to a finite direct product of matrix algebras over semigroup algebras of idempotent-free monoids?

Our main aim is to give some partial answer to these problems.

We proceed as follows: after citing known results, we prove that for an almost idempotent-free semigroup S satisfying the regularity condition, if K[S] is Azumaya, then $K[S]\cong M_n(K[G])$ where G is an idempotent-free monoid (Theorem 3.3). Based on this fact, we give a positive answer of Problem 1.3 when S satisfies the regularity condition (Corollary 3.6). In Sect. 4, we consider semigroup algebras of almost idempotent-free semigroups. We obtain a necessary and sufficient condition for the semigroup K[S] of an almost idempotent-free semigroup S satisfying the regularity condition and $min_J$ to be Azumaya (Theorems 4.4 and 4.6). It is proved that Conjecture 1.2 is true for the case when the idempotent-free semigroup satisfying $min_J$ (Theorem 4.10).

2 Preliminaries

Throughout this note, we use notations and terminologies from the monograph of Okniński [11] and the textbook of Clifford and Preston [4]. Other undefined terms can be found in the textbook [9].

2.1 Semigroups

Let S be a semigroup. Assume that I is an ideal of S. Form the set $S/I=(S\backslash I)\cup \{0\}$ and define a multiplication $\circ$ on S / I by the rule that

$$\begin{aligned} a\circ b=\left\{ \begin{array}{cl} ab &{} \quad \text { if } \,\, a,b,ab\in S\backslash I;\\ 0 &{} \quad \text { otherwise}, \end{array} \right. \end{aligned}$$

where ab is the product of a and b in the semigroup S. It is not difficult to check that $(S/I,\circ )$ is a semigroup. In what follows, we denote the above semigroup by S / I. In fact, S / I is the Rees factor semigroup of S modulo I (for detail, [4, p. 17]).

Also, S is called 0-simple if $S^2\not =0$ and S has at most two ideals: 0 and S. In what follows, we always use ${\mathcal {J}}$ to stand for the usual Green’s relation: $(a,b)\in {\mathcal {J}}$ if and only if there exist $x,y,u,v\in S^1$ such that $a=xby$ and $b=uav$. For $a\in S$, we use J(a) to denote the smallest ideal $S^1aS^1$ of S containing a, and $J_a$ to denote the ${\mathcal {J}}$-class of S containing a. We shall call the ${\mathcal {J}}$-class containing regular elements the regular ${\mathcal {J}}$-class of S. Define

$$\begin{aligned} J_a\le J_b \quad \text { if }\,\,\,J(a)\subseteq J(b). \end{aligned}$$

It is evident that $\le$ is a partial order on the set $S/\!{\mathcal {J}}$. When $J_a\le J_b$ and $J_a\not =J_b$, we shall denote $J_a<J_b$. As in Howie [9], we say that S satisfies the condition $min_J$ if the partially ordered set $S/\!{\mathcal {J}}$ satisfies the minimal condition. Note that $I(a)=\{x\in J(a): J_x< J_a\}$ is an ideal of S. The Rees factor semigroup $J(a)/I(a)=J_a\cup \{0\}$ of J(a) modulo I(a) is either a 0-simple semigroup or a null semigroup. For convenience, we denote the zero element by $0.\, J(a)/I(a)$ is called the principal factor of S determined by a. So, any principal factor is either a 0-simple semigroup or a null semigroup.

Let n be an integer. We say that S has the property ${\mathfrak {P}}_n$ if for any $x_1,x_2,\ldots ,x_n\in S$, there exists a nontrivial permutation $\sigma$ in the symmetric group ${\mathfrak {S}}_n$ such that $x_1x_2\ldots x_n=x_{\sigma (1)}x_{\sigma (2)}\ldots x_{\sigma (n)}$. It is easy to see that any subsemigroup and any homomorphic image of a semigroup having the property ${\mathfrak {P}}_n$ have the property ${\mathfrak {P}}_n$. For simplicity, we say that S has the permutational property if some ${\mathfrak {P}}_n, n\ge 2$, is satisfied in S. In [8], Domanov proved the following result (also see [11, Theorem 17, p. 229]):

Lemma 2.1

If T is a 0-simple semigroup satisfying the permutational property, then T is a completely 0-simple semigroup.

2.2 Semigroup Algebras

We always assume that K is a field and S a semigroup. We denote by K[S] the semigroup algebra of S over K. In general, if I is a subset of S, then K[I] denotes the set of K-linear combinations of elements in I, that is, K[I] is a vector space with I as a basis. So each element of K[I] is a finite summation of the form $\sum _{x\in I}r_xx, r_x\in K,x \in I$. In particular, if $I_1$ and $I_2$ are subsets of S, then $K[I_1\cap I_2] =K[I_1]\cap K[I_2]$. If S is a semigroup with zero $\theta$, then $K[\theta ]$ is an ideal of K[S], and we define $K_0[S]=K[S]/K[\theta ]$. This K-algebra $K_0[S]$ is called the contracted semigroup algebra of S over K. If S has no zero, then we have $K_0[S]=K[S]$. Clearly, an element a of $K_0[S]$ is a finite linear combination $a= \sum r_ss$ of elements $s\in S\backslash \{\theta \}$. The support of $a\in K_0[S]$, denoted by supp(a), is the set $\{s \in S\backslash \{\theta \}\mid r_s\not =0\}$.

Lemma 2.2

[11, Proposition 1, p. 221] Assume that K[S] satisfies a polynomial identity of degree n. Then S has the property ${\mathfrak {P}}_n$.

We now recall some known results on Azumaya algebras (for example, see [11, Lemma 1, p. 313]).

Lemma 2.3

Assume that ${\mathfrak {A}}$ is an Azumaya algebra, and denote by $Z({\mathfrak {A}})$ the center of ${\mathfrak {A}}$. Then

• (1) For every ideal ${\mathfrak {J}}$ of ${\mathfrak {A}}$ and ${\mathfrak {I}}$ of $Z({\mathfrak {A}})$, we have $({\mathfrak {J}}\cap Z({\mathfrak {A}})){\mathfrak {A}}={\mathfrak {J}}$ and ${\mathfrak {IA}}\cap Z({\mathfrak {A}})={\mathfrak {I}}$.

• (2) For every ideal ${\mathfrak {J}}$ of ${\mathfrak {A}}, Z({\mathfrak {A}}/{\mathfrak {J}})=(Z({\mathfrak {A}})+{\mathfrak {J}})/{\mathfrak {J}}$ and ${\mathfrak {A}}/{\mathfrak {J}}$ is an Azumaya algebra.

• (3) ${\mathfrak {A}}$ is a finitely generated projective module over its center $Z({\mathfrak {A}})$, and $Z({\mathfrak {A}})$ is a direct summand of this module.

In [1], Adjamagbo pointed out that

Lemma 2.4

[1, Recall 1.1 (8), p. 92] For any commutative algebra ${\mathfrak {A}}$ and commutative ${\mathfrak {A}}$-algebra ${\mathfrak {C}}$, if ${\mathfrak {B}}$ is separable over ${\mathfrak {A}}$, then $\mathfrak {B}\otimes _{{\mathfrak {A}}}{\mathfrak {C}}$ is separable over ${\mathfrak {C}}$. Conversely, if $\mathfrak {B}\otimes _{{\mathfrak {A}}}{\mathfrak {C}}$ is separable over ${\mathfrak {C}}$ and if, in addition, ${\mathfrak {B}}$ is a finitely generated ${\mathfrak {A}}$-module or ${\mathfrak {A}}$-algebra and ${\mathfrak {C}}$ a faithfully flat ${\mathfrak {A}}$-module, then ${\mathfrak {B}}$ is separable over ${\mathfrak {A}}$.

The following lemma follows immediately from [11, Proposition 13, p. 323].

Lemma 2.5

Let K[S] be an Azumaya algebra and $t\in S$. If the principal factor of S determined by t is 0-simple, then K[StS] is a ring direct summand of K[S].

By the proof of [11, Proposition 13, p. 323], Lemma 2.5 is true for $K_0[S]$.

The following observation is trivial.

Lemma 2.6

Let S be a semigroup. Then K[S] is an Azumaya algebra if and only if $K_0[S]$ is an Azumaya algebra.

We need the following well-known result (see [18, Proposition 25B.8] or [6]).

Lemma 2.7

Let S be a monoid. Then $M_n(K_0[S])$ is Azumaya if and only if $K_0[S]$ is Azumaya.

3 The Regularity Condition

A semigroup is said to satisfy the regularity condition if all of its regular elements form a subsemigroup. Obviously, cancellative monoids satisfy the regularity condition. In this section, we study when semigroup algebras of almost idempotent-free semigroups satisfying the regularity are Azumaya algebras.

Lemma 3.1

Let S be an almost idempotent-free semigroup. If K[S] is an Azumaya algebra, then the following statements are true:

• (1) For any $x\in S, x$ is not regular in S if and only if J(x) / I(x) is a null semigroup. Moreover, every element of the ${\mathcal {J}}$-class of S containing a nonzero regular element is regular.

• (2) The nonzero regular ${\mathcal {J}}$-class is the greatest element in the set $S/\!{\mathcal {J}}$ with the above order $\le$. Moreover, $V:=\big (S\backslash \mathrm{Reg}(S)\big )\cup \{0\}$ is an ideal of S.

• (3) S / V is a completely 0-simple semigroup. Moreover, for any nonzero regular elements a, b of $S, a{\mathcal {D}}b$.

Proof

• (1) Assume that x is not regular in S. Then J(x) / I(x) is a null semigroup; if not, then J(x) / I(x) is 0-simple, but since K[S] is Azumaya, now K[S] is a PI-algebra, so by Lemmas 2.1 and 2.2, J(x) / I(x) is a completely 0-simple semigroup. So, the assertions follow.

• (2) By hypothesis, K[S] has an identity. Let e be the identity of K[S] and J a maximum element of the set $A:=\{J_a:a\in \mathrm{supp}(e)\}$. If $b\in J\cap \mathrm{supp}(e)$, then by $be=b, bu=b$ for some $u\in \mathrm{supp}(e)$. Since $J=J_b=J_{bu}\le J_u$ and by the maximality of J, we have $J=J_u$. That is, $bu,u\in J$. This shows that $J(b)/I(b)=J\cup \{0\}$ is 0-simple. Thus by Lemmas 2.1 and 2.2, J(b) / I(b) is a completely 0-simple semigroup, so b is regular in S, and therefore J is a nonzero regular ${\mathcal {J}}$-class of S. For any $w\in S$, since $we=w$, there exists $z\in \mathrm{supp}(e)$ such that $w=wz$. It follows that $J_w\le J_z\le J$ because J is the greatest element of A. Note that S has only a nonzero regular ${\mathcal {J}}$-class. Therefore, the nonzero regular ${\mathcal {J}}$-class is the greatest element in the set $S/\!{\mathcal {J}}$. For any $a,b\in S$, if $ab\in \mathrm{Reg}(S)\backslash \{0\}$, then $J_{ab}\le J_a,J_b$ and $J_{ab}=J_a=J_b$ since the nonzero regular ${\mathcal {J}}$-class is maximal, so $a,b\in \mathrm{Reg}(S)\backslash \{0\}$, whence V is an ideal of S.

• (3) Because S is an almost idempotent-free semigroup and by $(2), S/V=J_a\cup \{0\}$ where a is regular. Since a is regular, we know that J(a) / I(a) is 0-simple, that is, S / V is a 0-simple semigroup. On the other hand, since K[S] is Azumaya, S / V satisfies the permutational property. Now by Lemma 2.1, S / V is a completely 0-simple semigroup. The rest is trivial. $\square$

Proposition 3.2

Let S be an almost idempotent-free semigroup. If K[S] is an Azumaya algebra, then S has finitely many idempotents.

Proof

By Lemma 3.1 (2), K[V] is an ideal of K[S], so that by Lemma 2.3 $(2), K_0[S/V]\cong K[S]/K[V]$ is Azumaya. It follows that $K_0[S/V]$ has an identity. Now by [11, Proposition 2.5, p. 59], S / V has finitely many ${\mathcal {L}}$-classes and ${\mathcal {R}}$-classes. Therefore, as S / V is a completely 0-simple semigroup, S has finitely many idempotents. $\square$

Theorem 3.3

Let S be an almost idempotent-free semigroup. If S satisfies the regularity condition, then $K_0[S]$ is an Azumaya algebra if and only if $K_0[S]\cong M_n(K_0[G])$ where G is an idempotent-free monoid and $K_0[G]$ is an Azumaya algebra.

Proof

Note that $K_0[S]=K_0[S^0]$. So, we may assume that S has zero element. By Lemma 2.7, we need only to verify the necessity. Now, assume that $K_0[S]$ is an Azumaya algebra. Then $K_0[S]$ has an identity, say E. Let $E=F+D$ with $\mathrm{supp}(F)\subseteq \mathrm{Reg}(S)$ and $\mathrm{supp}(D)\subseteq V$ where $V=(S\backslash \mathrm{Reg}(S))\cup \{0\}$. For any $a\in \mathrm{Reg}(S)$, we have $a=aE=aF+aD$ and $a=Ea=Fa+Da$. By Lemma 3.1(2), V is an ideal of S, and $\mathrm{supp}(aD),\mathrm{supp}(Da)\subseteq V$. Thus $a=Fa=aF$, in other words, F is an identity of $K_0[\mathrm{Reg}(S)]$.

Compute

$$\begin{aligned} F=FE=F^2+FD=EF=F^2+DF. \end{aligned}$$

But $DF,FD\in K[V]$ and $F^2\in K[\mathrm{Reg}(S)]$, so $F^2=F$ and $FD=0=DF$. Since

$$\begin{aligned} F+D= & {} E=E^2=(F+D)^2=F^2+FD+DF+D^2\\= & {} F^2+D^2 \end{aligned}$$

and by $D^2\in K[V]$ and $F^2\in K[\mathrm{Reg}(S)]$, we observe that $D=D^2$. Let $x\in \mathrm{supp}(D)$ such that $J_x$ is the maximum element of the set $\big \{J_u:u\in \mathrm{supp}(D)\big \}$. The fact that $D=D^2$ derives that $x=yz$ for some $y,z\in \mathrm{supp}(D)$. It follows that $J_x\le J_y,J_z$. Thus $J_x=J_y=J_z$ by the maximality of $J_x$. Now, $x=yz$ can show that J(x) / I(x) is 0-simple, and further by Lemma 3.1(1), x is a regular element, contradicting to the fact that $\mathrm{supp}(D)\subseteq V$. Consequently, $D=0$ and whence $E=F$ and is the identity of $K_0[S]$.

By Lemma 3.1 (3), S / V is isomorphic to some Rees matrix semigroup $\mathcal {M}^0(H,I,\Lambda ;P)$ over a group H. Again by Lemma 3.1 $(2), S/V\cong \mathrm{Reg}(S)$, so that $K_0[S/V]=K_0[\mathrm{Reg}(S)]$. For convenience, we identify $\mathrm{Reg}(S)$ with $\mathcal {M}^0(H,I,\Lambda ;P)$. By [11, Lemma 1, p. 48], the mapping $\psi$ linearly spanned by the the mapping defined by

$$\begin{aligned} K_0\big [\mathrm{Reg}(S)\big ]\rightarrow {\mathfrak {M}}\big (K[H],I,\Lambda ;P\big ); (a,i,m)\mapsto (a_{jn}) \end{aligned}$$

is an algebra isomorphism, where ${\mathfrak {M}}\big (K[H],I,\Lambda ;P\big )$ is an algebra of matrix type and $(a_{jn})$ the $I\times \Lambda$ matrix with entry $a_{jn}=a$ if $j=i,n=m$ and, otherwise, $a_{jn}=0$. Since, by the forgoing proof, $K_0\big [\mathrm{Reg}(S)\big ]$ has an identity, and by [11, Proposition 25, p. 59], this shows that $|I|=|\Lambda |=n<+\infty$ and P is an invertible matrix in $M_n\big (K[H]\big )$. It is not difficult to check that $P^{-1}$ is the identity of ${\mathfrak {M}}\big (K[H],I,\Lambda ;P\big )$ and the mapping defined by $\varphi : \;A\mapsto AP$ is an isomorphism of ${\mathfrak {M}}\big (K[H],I,\Lambda ;P\big )$ onto $M_n(K[H])$.

Denote by $\varepsilon _{i}$ the $n\times n$ matrix with unity of K[H] at the row i, column i position and the zero 0 of K[H] in all other entries. So, $F=\sum _{i=1}^nf_i$ where $f_i=\psi ^{-1}\varphi ^{-1}\left( \varepsilon _{i}\right)$. Put

$$\begin{aligned} M_{im}=\big \{(a,i,m): a\in H\big \}. \end{aligned}$$

It is not difficult to know that each $f_i$ is an idempotent, supp $(f_i)\subseteq M_{ii}$ and $f_if_m\not =0$ only if $i=m$.

Lemma 3.4

Let S satisfy the conditions of Theorem 3.3. Then $K_0[S]=\bigoplus _{i=1}^ne_iK_0[S]$, where $e_i=(p_{m_i,i}^{-1}, i, m_i) (m_i\in \Lambda )$ and $e_iK_0[S]\cong e_jK_0[S]$ for any i, j.

Proof

By the properties of Rees matrix semigroups, $e_ix=x=xe_i$ for all $x\in M_{ii}$. It follows that

$$\begin{aligned} f_i\cdot K_0[S]\subseteq \sum _{u\in \mathrm{supp}\; (f_i)}u\cdot K_0[S]\subseteq e_i\cdot K_0[S]. \end{aligned}$$

Because $\mathrm{supp}\; (f_je_i)\subseteq M_{ji}$, we have

$$\begin{aligned} \mathrm{supp}\; \left( \sum _{j=1}^{i-1}f_je_i+\sum _{j=i+1}^nf_je_i\right) \; \bigcap M_{ii}=\emptyset \end{aligned}$$

and $\mathrm{supp}\; (f_ie_i)\subseteq M_{ii}$. But

$$\begin{aligned} e_i=f_ie_i+\sum _{j=1}^{i-1}f_je_i+\sum _{j=i+1}^nf_je_i, \end{aligned}$$

now $e_i=f_ie_i.$ Therefore, $e_i\cdot K_0[S]\subseteq f_i\cdot K_0[S]$ and whence $e_i\cdot K_0[S]= f_i\cdot K_0[S]$.

Note that $F=\sum _{i=1}^nf_i$ is the identity of $K_0[S]$ and $f_i$’s are orthogonal. We observe that $K_0[S]=\bigoplus _{i=1}^nf_i\cdot K_0[S]=\bigoplus _{i=1}^ne_i\cdot K_0[S]$. In addition, $e_i$’s are all nonzero, further by Lemma 3.1 $(3), e_i$s are related by ${\mathcal {D}}$. Therefore, by [10, Proposition (21.20), p. 315], $e_iK_0[S]\cong e_jK_0[S]$ for any i, j. $\square$

Now, we have

$$\begin{aligned} K_0[S]= & {} \bigoplus \nolimits _{i=1}^ne_i\cdot K_0[S]\cong End_{K_0[S]}(n e_1\cdot K_0[S])\cong M_n(e_1K_0[S]e_1)\\\cong & {} M_n(K_0[e_1Se_1]). \end{aligned}$$

On the other hand, since S is an almost idempotent-free semigroup, it is easy to see that $E(e_1Se_1)=E(e_1\mathrm{Reg}(S)e_1)$. But $S/V\cong \mathrm{Reg}(S)$ is a completely 0-simple semigroup, we have $E(e_1\mathrm{Reg}(S)e_1)=\{e_1,0\}$. However, $e_1Se_1$ is an idempotent-free monoid.

We have now proved that $K_0[S]\cong M_n(K_0[e_1Se_1])$, so that $M_n(Z(K_0[e_1Se_1]))$ is an Azumaya algebra. By Lemma 2.7, $K_0[e_1Se_1]$ is an Azumaya algebra. The proof is finished. $\square$

Let us turn back to the proof of Theorem 3.3. If $S\backslash V$ is a subsemigroup of S, then P has no zero entries. But P is invertible in $M_n(K[H])$, so the cardinality of the sets I and $\Lambda$ must be equal to 1. In other words, $n=1$. Thus S is an idempotent-free monoid. Note that $S\backslash V$ is a subsemigroup of S is equivalent to that the set of nonzero regular elements of S forms a subsemigroup of S, we have the following corollary.

Corollary 3.5

Let S be an almost idempotent-free semigroup whose set of nonzero regular elements forms a subsemigroup. If K[S] is Azumaya, then S is an idempotent-free monoid.

By applying the result [11, Corollary 17, p. 325], we have that for any finite semigroup $S, K_0[S]$ is an Azumaya algebra if and only if $K_0[S]\cong \oplus _{i=1}^mK_0[S_i]$ where each $S_i$ is an almost idempotent-free semigroup and $K_0[S_i]$ an Azumaya algebra. Furthermore, by Theorem 3.2, the following corollary is immediate. This corollary confirms Problem 1.3 when the semigroup satisfies the regularity condition.

Corollary 3.6

Let S be a finite semigroup. If S satisfies the regularity condition, then $K_0[S]$ is an Azumaya algebra if and only if $K_0[S]\cong \bigoplus _{i=1}^mM_{n_i}(K_0[G_i])$ where each $G_i$ is an idempotent-free monoid and $K_0[G_i]$ an Azumaya algebra.

4 The Condition $min_J$

The aim of this section is to determine when semigroup algebras of almost idempotent-free semigroups satisfying the condition $min_J$ are Azumaya algebras.

Lemma 4.1

Let S be an almost idempotent-free semigroup satisfying the condition $min_J$. Assume that K[S] is an Azumaya algebra. If I and J are ${\mathcal {J}}$-classes of S that are not regular, then for any $a\in I,b\in J$, we have $J_{ab}<I,J$ and $J_{ba}<I,J$.

Proof

For any $a\in I,b\in J$, we have $J_{ab}\le J_b=J$. If $J_{ab}=J$, then there exist $x,y\in S^1$ such that $b=xaby$. Thus $b=(xa)^nby^n$ for all positive integer n. Obviously, $(xa)^n\not =0$. Note that

$$\begin{aligned} \cdots \le J_{(xa)^3}\le J_{(xa)^2}\le J_{xa}. \end{aligned}$$

But S satisfies the condition $min_J$, so there exists positive integer m such that

$$\begin{aligned} J_{(xa)^m}=J_{(xa)^{m+1}}=\cdots . \end{aligned}$$

It follows that $(xa)^m(xa)^m\in J_{(xa)^m}$, in other words, $J_{(xa)^m}$ is 0-simple. Thus by Lemma 3.1 $(1), (xa)^m$ is regular, whence by Lemma 3.1(2), I is the unique nonzero regular ${\mathcal {J}}$-class of S since $J_{(xa)^m}\le J_a=I$, giving $J_{(xa)^m}=I$. This is a contradiction. Consequently, $J_{ab}<J$; similarly $J_{ab}<I$. The rest can be similarly proved. $\square$

Lemma 4.2

Let S be an almost idempotent-free semigroup satisfying the condition $min_J$. If K[S] is an Azumaya algebra, then for any $a\in S$, either a is regular or $a^n=0$ for some positive integer n if S has zero element 0.

Proof

Without loss of generality, we assume that S has zero element 0. Now, let a be not a regular element of S. Obviously,

$$\begin{aligned} J_a\ge J_{a^2}\ge \cdots \ge J_{a^n}\ge \cdots . \end{aligned}$$

But S satisfies the condition $min_J$, so there exists positive integer m such that

$$\begin{aligned} J_{a^m}=J_{a^{m+1}}=\cdots . \end{aligned}$$

It follows that $J_{a^m}=J_{a^{2m}}$, whence $a^m\circ a^m\in J_{a^m}$. Thus $a^m=0$ or $J(a^m)/I(a^m)$ is 0-simple. If the second case holds, then by Lemma 3.1 $(1), a^m$ is regular. Note that $J_{a^m}\le J_a$. By Lemma 3.1 $(2), J_{a^m}=J_a$ and whence a is regular, contrary to the hypothesis. Therefore, $a^m=0$ and the result follows. $\square$

Recall that a semigroup S is called an ideal extension of nil semigroups by a completely 0-simple semigroup if S has a nil ideal I such that S / I is a completely 0-simple semigroup. The following corollary is immediate from Lemmas 3.1 and 4.2.

Corollary 4.3

Let S be an almost idempotent-free semigroup satisfying $min_J$. If $K_0[S]$ is Azumaya, then S is an ideal extension of nil semigroups by a completely 0-simple semigroup.

Based on Lemma 4.2, we have the following theorem.

Theorem 4.4

Let S be an almost idempotent-free semigroup satisfying the condition $min_J$, and assume that S has no zero elements. If K[S] is an Azumaya algebra, then S is a group.

Proof

Assume that K[S] is an Azumaya algebra. By Lemma 4.2, all elements of S are regular, in other words, S is a regular semigroup. By Corollary 3.5, S is an idempotent-free monoid. Thus S is a group. $\square$

We shall simply denote

$$\begin{aligned} Z\big (K[S]\big )K\big [\mathrm{Reg}(S)\big ]=\left\{ \sum _{i=1}^mz_iy_i: z_i\in Z\big (K[S]\big ),y_i\in K\big [\mathrm{Reg}(S)\big ], m\in N\right\} , \end{aligned}$$

where N is the set of positive integers.

Lemma 4.5

Let S be an almost idempotent-free semigroup satisfying the condition $min_J$. If K[S] is an Azumaya algebra, then $K[S]=Z\big (K[S]\big )K\big [\mathrm{Reg}(S)\big ]$.

Proof

Assume that K[S] is an Azumaya algebra. By Theorem 4.4, the theorem is clearly true for the case if S has no zero elements. So, we now let S have a zero element. Assume, on the contrary, that there exists $\alpha \notin Z\big (K[S]\big )K\big [\mathrm{Reg}(S)\big ]$. It follows that $a_1\notin Z\big (K[S]\big )K\big [\mathrm{Reg}(S)\big ]$ for some $a_1\in \mathrm{supp}(\alpha )$. Obviously, $a_1\in S\backslash \mathrm{Reg}(S)$. By Lemma 2.3(1), we have

$$\begin{aligned} \Big (K\big [J(a_1)\big ]\cap Z\big (K[S]\big )\Big )K[S]=K\big [J(a_1)\big ] \end{aligned}$$

(1)

and $a_1=\alpha _1 (\beta _1+\beta '_1)=\alpha _1\beta _1+\alpha _1\beta '_1$ for some $\alpha _1\in K[J(a_1)]\cap Z(K[S]), \beta _1\in K[\mathrm{Reg}(S)]$ and $\beta '_1\in K[V]$. But $a_1\notin Z(K[S])K[\mathrm{Reg}(S)]$, so $\alpha _1\beta '_1\notin Z(K[S]) K[\mathrm{Reg}(S)]$ and $\alpha _1\beta '_1\not =0$. Since $J(a_1)=J_{a_1}\cup I(a_1)$, we know that $J_{a_1}$ is the greatest element of the set $\{J_x:x\in J(a_1)\}$, and whence $J_{a_1}$ is bigger than the greatest element of the set

$$\begin{aligned} B:=\Big \{J_x: x\in \mathrm{supp}(u),u\in K\big [J(a_1)\big ]\cap Z\big (K[S]\big )\Big \}. \end{aligned}$$

In particular, $J_{a_1}\ge J_x$ for any $x\in \mathrm{supp}(\alpha _1)$. Now, by Lemma 4.1, $J_{a_1}> J_y$ for any $y\in \mathrm{supp}(\alpha _1\beta '_1)$. Because $\alpha _1\beta '_1\notin Z\big (K[S]\big )K\big [\mathrm{Reg}(S)\big ]$, we have $a_2\in \mathrm{supp}(\alpha _1\beta '_1)$ such that $a_2\notin Z(K[S])K[\mathrm{Reg}(S)]$. By the forgoing proof, $J_{a_1}>J_{a_2}$.

By applying the same arguments to $a_2$, we have $a_3$ such that $a_3\notin Z(K[S])K[\mathrm{Reg}(S)]$ and $J_{a_2}>J_{a_3}$. Continuing this process, we may obtain the ${\mathcal {J}}$-classes:

$$\begin{aligned} J_{a_1}>J_{a_2}>\cdots >J_{a_n}>\cdots , \end{aligned}$$

contrary to the hypothesis that S satisfies the condition $min_J$. Consequently, $K[S]=Z\big (K[S]\big )K\big [\mathrm{Reg}(S)\big ]$. $\square$

We now arrive at the main result of this section.

Theorem 4.6

Let S be an almost idempotent-free semigroup satisfying $min_J$. If, in addition, S satisfies the regularity condition, then K[S] is an Azumaya algebra if and only if $K[S]=Z(K[S])K[\mathrm{Reg}(S)]$ and $K\big [\mathrm{Reg}(S)\big ]$ is an Azumaya algebra.

Proof

Assume that K[S] is an Azumaya algebra. By Lemma 4.5, we need only to verify that $K\big [\mathrm{Reg}(S)\big ]$ is an Azumaya algebra. In fact, by Lemma 3.1 (2), V is an ideal of S, so by Lemma 3.1 $(2), K_0[\mathrm{Reg}(S)]\cong K_0[S/V]\cong K[S]/K[V]$ and further is an Azumaya algebra by Lemma 2.3. However, by Lemma 2.6, $K[\mathrm{Reg}(S)]$ is an Azumaya algebra.

Conversely, suppose that $K[S]=Z\big (K[S]\big )K\big [\mathrm{Reg}(S)\big ]$ and $K\big [\mathrm{Reg}(S)\big ]$ is an Azumaya algebra. Since $K[S]=Z\big (K[S]\big )K\big [\mathrm{Reg}(S)\big ]$, we know that the center $Z\big (K[S]\big )$ of K[S] is equal to

$$\begin{aligned} \Big \{x\in K[S]: xu=ux \quad \text { for all }u\in K\big [\mathrm{Reg}(S)\big ]\Big \}. \end{aligned}$$

This means that the center $Z\big (K[\mathrm{Reg}(S)]\big )$ of $K\big [\mathrm{Reg}(S)\big ]$ is contained in $Z\big (K[S]\big )$. By Lemma 2.4, we observe that $Z\big (K[S]\big )\otimes _{Z\big (K\big [\mathrm{Reg}(S)\big ]\big )}K\big [\mathrm{Reg}(S)\big ]$ is an Azumaya algebra. It is easy to see that K[S] is a homomorphic image of $Z(K[S])\otimes _{Z(K[\mathrm{Reg}(S)])}K[\mathrm{Reg}(S)]$ since $K[S]=Z(K[S])K[\mathrm{Reg}(S)]$. Now by Lemma 2.3 (2), K[S] is an Azumaya algebra. $\square$

It is well known that finite semigroups satisfy the condition $min_J$. Obviously, any infinite group satisfies the condition $min_J$. So, not all semigroups satisfying the condition $min_J$ are finite. The next example illustrates that there exist Azumaya semigroup algebras of almost idempotent-free semigroups satisfying the condition $min_J$ but not finite.

Example 4.7

Let U be a null semigroup with zero 0, and G a group with identity 1 and such that K[G] is an Azumaya algebra. Assume that S is the disjoint union of U and G. Define a multiplication $*$ by:

$$\begin{aligned} x*y=\left\{ \begin{array}{cl} x&{} \quad \text { if}\quad y\in G\quad \hbox {but}\quad x\in U\\ y&{} \quad \text { if}\quad x\in G\quad \hbox {but}\quad y\in U\\ xy&{} \quad \text { otherwise,} \end{array} \right. \end{aligned}$$

where xy is the product of x and y in the semigroup U or the group G. By computation, $(S,*)$ is a monoid with identity 1. It is easy to check that the Green’s relation ${\mathcal {J}}$ on S is equal to $(G\times G)\cup \Delta _U$ where $\Delta _U$ is the identity relation on U. This shows that in the semigroup S, maximal chains of ${\mathcal {J}}$-classes of S have the form: $J_0<J_x<J_g,$ where $x\in U\backslash \{0\}, g\in G$. So, S satisfies $min_J$. On the other hand, it is easy to see that $Z\big (K[S]\big )=K[U]+Z\big (K[G]\big )$ and whence $K[S]=Z\big (K[S]\big )K[G]$ since $1\in Z\big (K[S]\big )$. By Theorem 4.6, K[S] is an Azumaya algebra. When U is infinite, S is clear infinite.

The following example illustrates the condition that $min_J$ is not necessary for a semigroup algebra K[S] to be an Azumaya algebra.

Example 4.8

Let G be a group such that K[G] is Azumaya. Let U be the $\omega$-chain $\{e_0,e_1, e_2,\ldots \}$ with

$$\begin{aligned} e_0>e_1>e_2>\ldots \end{aligned}$$

(for detail, see [9, Example 4.6, p. 144]). The semigroup S constructed as in Example 4.7 has the ${\mathcal {J}}$-class chain:

$$\begin{aligned} J_1>J_{e_0}>J_{e_1}>J_{e_2}>\cdots . \end{aligned}$$

This shows that S does not satisfy the condition $min_J$. Note that U is in the center of S. Thus $Z\big (K[S]\big )=Z\big (K[G]\big )+K[U]$, so that $K[S]=Z\big (K[S]\big )K[G]$. It follows that K[S] is a homomorphic image of $K[G]\otimes _{Z\big (K[G]\big )}Z\big (K[S]\big )$. But $K[G]\otimes _{Z\big (K[G]\big )}Z\big (K[S]\big )$ is Azumaya (by Lemma 2.4). Therefore, K[S] is an Azumaya algebra.

The remainder of this section is devoted to semigroup algebras of idempotent-free semigroups.

Lemma 4.9

Let S be an idempotent-free semigroup. If K[S] is an Azumaya algebra, then S is a monoid satisfying the regularity condition and $\mathrm{Reg}(S)\backslash \{0\}$ is a subgroup of S. Moreover, if S has no zero elements, then S is a group.

Proof

Because S is an idempotent-free semigroup and by Lemma 3.1 (3), S / V is a completely 0-simple semigroup without idempotents except possibly identity and zero element. It follows that S / V is a 0-group (that is, a group adjoining a zero). This shows that $\mathrm{Reg}(S)\backslash \{0\}$ is a subgroup of S, and whence S satisfies the regularity condition. Now, by the proof of Theorem 3.3, the identity of K[S] coincides with the one of $K[\mathrm{Reg}(S)]$, so that S is a monoid. The rest is trivial. $\square$

By Lemma 4.9 and Theorem 4.6, the following theorem is immediate. This theorem answers particularly Conjecture 1.2.

Theorem 4.10

Let S be an idempotent-free monoid with the group of units U. If S satisfies the condition $min_J$, then K[S] is an Azumaya algebra if and only if $K[S]=Z\big (K[S]\big )K[U]$ and K[U] is an Azumaya algebra.

It is clear to see that cancellative monoids are idempotent-free semigroups. By Theorem 4.4, we have the following corollary.

Corollary 4.11

Let S be a cancellative monoid satisfying the condition $min_J$. If K[S] is an Azumaya algebra, then S is a group.

Proof

By Theorem 4.4, S is isomorphic to some Rees matrix semigroup over a group, and further a regular semigroup. But any regular cancellative monoid is a group, now S is a group. $\square$