Duo Ring Property Restricted to Groups of Units

Abstract

Abstract. We study the structure of right duo ring property when it is restricted within the group of units, and introduce the concept of right unit-duo. This newly introduced property is first observed to be not left-right symmetric, and we examine several conditions to ensure the symmetry. Right unit-duo rings are next proved to be Abelian, by help of which the class of noncommutative right unit-duo rings of minimal order is completely determined up to isomorphism. We also investigate some properties of right unit-duo rings which are concerned with annihilating conditions.Throughout this paper all rings are associative with identity unless otherwise stated. Let R be a ring. X(R) denotes the set of all nonzero nonunits in R, and G(R) denotes the group of all units in R., and N (R) denote the Jacobson radical, the set of all idempotents, and the set of all nilpotent elements in R, respectively. |S| denotes the cardinality of a subset S of R. Write R * = R\{0}. Z (Z n ) denotes the ring of integers (modulo n). Q denotes the field of rational numbers. GF (p n ) denotes the Galois field of order p n .Denote the n by n full (resp., upper triangular) matrix ring over R by Mat n (R) (resp., U n (R)) and use e ij for the matrix with (i, j)-entry 1 and elsewhere 0. Following the literature, we write D n (R) = {(a ij ) ∈ U n (R) | all diagonal entries are equal } and V n (R) = {(a ij ) ∈ D n (R) | a 1k = a 2(k+1) = · · · = a hn for h = 1, 2, . . . , n − 1 and k = 2, . . . , n}.Due to Feller [7], a ring is called right (resp. left) duo if every right (resp. left) ideal is an ideal; a ring is called duo if it is both right and left duo. It is easily shown that idempotents of right duo rings are central. There are very useful results for duo rings in [3,16,24].

Structures Concerning Group of Units

“…Following Han et al [14], a ring is called right unit-duo if [a] ℓ ⊆ [a] r for every a ∈ R. Left unit-duo rings are defined similarly. A ring is unit-duo if it is both We consider next the right unit-duo ring property on the powers of elements.…”

“…for all a i ∈ K and g i ∈ G. We have also that KG is left unit-duo if and only if it is right unit-duo by [14,Theorem 1.3(4)]. Recall that Q 8 denotes the quaternion group.…”

“…Example 2.8. We refer to the arguments in [14,Example 1.7] and [22, Theorem 1.3.5, Corollary 2.1.14, and Theorem 2.1.15]. Let Q x, y be the free algebra with noncommuting indetermiantes x, y over Q.…”

Unit-Duo Rings and Related Graphs of Zero Divisors

“…In [8], it was shown that any right (left) unit-duo ring R is abelian (i.e., every idempotent in R is central). In [11], it was also shown that if a ring R has a finite number of equivalence classes under ≃ ℓ , then R is an artinian ring with J(R) n+1 = 0 where n is the number of classes under ≃ ℓ .…”

“…∼ r ). A ring R is called unit-duo if [x] ℓ = [x] r for all x ∈ R (refer [8]). Any commutative ring and a finite direct product of division rings are unit-duo rings.…”