Notation and Terminology

Abstract

Any ring A in this book is associative with identity 1 (for emphasis this is sometimes written as 1_A). A subring has the same identity, and ring homomorphisms preserve 1s. The possibility that A= {0} is allowed. Obviously in this case, 1 = 0. If A ≠ {0}, then 1 ≠ 0; for otherwise, a = a·1 = a·0 = 0 for any a in A. If A ≠ {0} and ab = 0 implies that either a = 0 or b = 0, then A is a domain. If A ≠ {0}, and {0} and A are the only two-sided ideals of A, then A is simple. For a subset S of A, the centralizer of S in A is the subring Cen_A S of A defined by

$$ Ce{n_A}S = \left\{ {a \in A\left| {as = sa\,for\,al} \right.l\,s \in S} \right\} $$

.