REFLEXIVE PROPERTY ON IDEMPOTENTS

Abstract

The reflexive property for ideals was introduced by Mason and has important roles in noncommutative ring theory. In this note we study the structure of idempotents satisfying the reflexive property and introduce reflexive-idempotents-property (simply, RIP) as a generaliza- tion. It is proved that the RIP can go up to polynomial rings, power series rings, and Dorroh extensions. The structure of non-Abelian RIP rings of minimal order (with or without identity) is completely investi- gated. Throughout this paper all rings are associative with identity unless otherwise stated. Given a ring R the polynomial ring (resp., the power series ring) with an indeterminate x over R is denoted by R(x) (resp., R((x))). For any polynomial f(x) in R(x), let Cf(x) denote the set of all coefficients off(x). Denote the n by n full matrix ring over R by Matn(R) and the n by n upper triangular matrix ring over R by Un(R). Use Eij for the matrix with (i,j)-entry 1 and elsewhere 0. Let Id(R) be the set of all idempotent elements of R. Denote {a ∈ Un(R) | the diagonal entries of a are all equal} by Dn(R). Z and Zn denote the ring of integers and the ring of integers modulo n, respectively. GF(p n ) denotes the Galois field of order p n for a prime p and n ≥ 1. J(R) denotes the Jacobson radical of R. |S | denotes the cardinality of given a set S. Mason (18) introduced the reflexive property for ideals, and then this concept was generalized by Kim and Baik (9, 10) by defining idempotent reflexive right ideals and rings. A right ideal I of a ring R (possibly without identity) is called reflexive (18) if aRb ⊆ I implies bRa ⊆ I for a,b ∈ R. R is called reflexive if 0 is a reflexive ideal (i.e., aRb = 0 implies bRa = 0 for a,b ∈ R.) In (12), Kwak and Lee characterized aspects of the reflexive and one-sided idempotent reflexive properties, showing that the concept of idempotent reflexive ring is not left- right symmetric (12, Example 3.3). For a one-sided ideal I of a ring R, I is