STRUCTURE OF IDEMPOTENTS IN RINGS WITHOUT IDENTITY

Abstract

We study the structure of idempotents in polynomial rings, power series rings, concentrating in the case of rings without identity. In the procedure we introduce right Insertion-of-Idempotents-Property (simply, right IIP) and right Idempotent-Reversible (simply, right IR) as generalizations of Abelian rings. It is proved that these two ring proper- ties pass to power series rings and polynomial rings. It is also shown that �-regular rings are strongly �-regular when they are right IIP or right IR. Next the noncommutative right IR rings, right IIP rings, and Abelian rings of minimal order are completely determined up to isomorphism. These results lead to methods to construct such kinds of noncommuta- tive rings appropriate for the situations occurred naturally in studying standard ring theoretic properties. 1. Definitions and notations Throughout this paper, R denotes an associative ring without identity, unless otherwise stated. Denote the n by n full (resp., upper triangular) matrix ring over R by Matn(R) (resp., Un(R)). We let eij denote the usual matrix units with 1 in the (i,j)-position and zeros elsewhere, if the base ring has identity 1. Denote {(aij) ∈ Un(R) | the diagonal entries of (aij) are all equal} by Dn(R). Zn denotes the ring of integers modulo n. GF(p n ) denotes the Galois field of order p n . J(R) denotes the Jacobson radical of R. | | denotes the cardinality. The characteristic of R is denoted by charR, and |a| denotes the order of a ∈ R in the additive subgroup of R generated by a. R + means the additive Abelian group (R,+). The polynomial ring with an indeterminate x over R is denoted by R(x). While speaking about minimal ring in a certain class of rings, we refer to a ring with minimal order for rings in that class, due to Xue (21). The notation (S) stands for the two-sided ideal of R generated by ∅ 6 S ⊆ R, and we also write (a1,...,an) in place of (S) for simplicity when S = {a1,...,an}. A ring is called Abelian if every idempotent is central. The zero in a nil ring is the only idempotent and so every nil ring is Abelian. The class of Abelian