### MINIMAL NONCOMMUTATIVE REVERSIBLE AND REFLEXIVE RINGS

Byung-Ok Kim, Yang Lee
2011 Bulletin of the Korean Mathematical Society
The reflexiveness and reversibility were introduced by Mason and Cohn respectively. The structures of minimal reversible rings and minimal reflexive rings are completely determined. The term minimal means having smallest cardinality. Throughout this note all rings are associative with identity unless otherwise stated. Let R be a ring. The n by n full (resp. upper triangular) matrix ring over R is denoted by Mat n (R) (resp. U n (R)). Let J(R) denote the Jacobson radical of R. Z n denotes the
more » ... Z n denotes the ring of integers modulo n. According to Cohn [2], a ring R is called reversible if ab = 0 implies ba = 0 for a, b ∈ R. Anderson-Camillo [1] used the term ZC 2 for the reversibility. A ring is called reduced if it has no nonzero nilpotent elements. Reduced rings are reversible via a simple computation. Commutative rings are clearly reversible, but the converse need not hold since there exist many kinds of noncommutative reduced rings. Note also that there are many kinds of non-reduced commutative rings (e.g., Z k n for any k, n ≥ 2). A ring is called Abelian if every idempotent is central. Reversible rings are Abelian through a simple computation. Due to Mason [12], a ring R is called reflexive if aRb = 0 implies bRa = 0 for a, b ∈ R. Semiprime rings are reflexive by a simple computation. Reversible rings are clearly reflexive, but the converse need not hold since there exist many kinds of non-Abelian semiprime rings (e.g., Mat n (R) for a semiprime ring R and n ≥ 2). Note that reflexive rings need not be Abelian as can be seen by Mat n (R) (n ≥ 2) over a semiprime ring R. A ring R is called semilocal if R/J(R) is semisimple Artinian, and R is called semiperfect if R is semilocal and idempotents can be lifted modulo J(R). R is called local if R/J(R) is a division ring. Local rings are clearly semilocal, and one important case of semiperfect rings is when the Jacobson radical is nil