A ring [Formula: see text] is said to be clean if each element of can written as the sum a unit and an idempotent. weakly either or difference idempotent, feebly every text], where are orthogonal idempotents. Clearly, rings both them clean. In recent paper (J. Algebra Appl. 17 (2018) 1850111 (5 pp.)), McGoven characterized when group clean, distinct primes. this paper, we consider more general ...

Let R be an associative ring in which an identity element is not assumed. A right quotient ring of P is an overring 5 such that for each aQS there corresponds rQR such that arQR and ar 9*0. A theorem of R. E. Johnson [l ] states that R possesses a right quotient ring S which is a (von Neumann) regular ring if and only if P has vanishing right singular ideal. In this case P possesses a unique (u...

A ring R is defined to be semiabelian if every idempotent of R is either right semicentral or left semicentral. It is proved that the set N(R) of nilpotent elements in a π-regular ring R is an ideal of R if and only if R/J(R) is abelian, where J(R) is the Jacobson radical of R. It follows that a semiabelian ring R is π-regular if and only if N(R) is an ideal of R and R/N(R) is regular, which ex...

Let R be an hereditary Noetherian prime ring, let S be a "Dedekind closure" of R and let T be the category of finitely generated S-torsion R-modules. It is shown that for all i ~ 0, there is an exact sequence 0 -> Ki(T) -> Ki(R) -> Ki(S) ...... O. If i = 0, or R has finitely many idempotent ideals then this sequence splits. A notion of ''right ideal class group" is then introduced for hereditar...

Let mod kG be the stable category of finitely generated modular representations of a finite group G over a field k. We prove a Krull-Remak-Schmidt theorem for thick subcategories of mod kG. It is shown that every thick tensor-ideal C of mod kG (i.e. a thick subcategory which is a tensor ideal) has a (usually infinite) unique decomposition C = ∐ i∈I Ci into indecomposable thick tensor-ideals. Th...

A left almost semigroup (LA-semigroup) or an Abel-Grassmann’s groupoid (AG-groupoid) is investigated in several papers. In this paper we have discussed ideals in LA-semigroups. Specifically, we have shown that every ideal in an LA-semigroup S with left identity e is prime if and only if it is idempotent and the set of ideals of S is totally ordered under inclusion. We have shown that an ideal o...

Recall that a rng is a ring which is possibly non-unital. In this note, we address the problem whether every finitely generated idempotent rng (abbreviated as irng) is singly generated as an ideal. It is well-known that it is the case for a commutative irng. We prove here it is also the case for a free rng on finitely many idempotents and for a finite irng. A relation to the Wiegold problem for...

We introduce in this paper the concept of left WMC2 rings and concern ourselves with rings containing an injective maximal left ideal. Some known results for left idempotent reflexive rings and left HI rings can be extended to left WMC2 rings. As applications, we are able to give some new characterizations of regular left self-injective rings with nonzero socle and extend some known results on ...

We study the action of G = SL(2,R), viewed as a group definable in the structure M = (R,+,×), on its type space SG(M). We identify a minimal closed G-flow I, and an idempotent r ∈ I (with respect to the Ellis semigroup structure ∗ on SG(M)). We also show that the “ideal group” (r ∗ I, ∗) is nontrivial (in fact it will be the group with 2 elements), yielding a negative answer to a question of Ne...