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Let be a strictly totally ordered monoid and R an-weakly rigid ring, where is a monoid homomorphism. In this paper, we study the weakly pq-Bear property of the skew generalized power series ring. As a consequence, the weakly pq-Baer property of the skew power series ring and the skew Laurent power series ring are determined, where is a ring endomorphism of R.
A skew generalized power series ring R [[S, ω]] consists of all functions from a strictly ordered monoid S to a ring R whose support contains neither infinite descending chains nor infinite antichains, with pointwise addition, and with multiplication given by convolution twisted by an action ω of the monoid S on the ring R. Special cases of the skew generalized power series ring construction are skew polynomial rings, skew Laurent polynomial rings, skew power series rings, skew Laurent series rings, skew monoid rings, skew group rings, skew Mal’cev-Neumann series rings, the “untwisted” versions of all of these, and generalized power series rings. In this paper we obtain necessary and sufficient conditions on R, S, and ω such that t
A right module is extending if every submodule is essential in a direct summand of . In this note, we obtain a characterization of the right extending generalized triangular matrix rings. This answers a question which was raised in “E. Akalan, G. F. Birkenmeier and A. Tercan, Characterizations of extending modules and -extending generalized triangular matrix rings, Comm. Algebra 40 (2012) 1069–1085”.
Let be a unitary ring with an endomorphism and be the free monoid generated by with added, and be a factor of setting certain monomial in to , enough so that, for some natural number , . In this paper, we give a sufficient condition for a ring such that the skew monoid ring is quasi-Armendariz (By Hirano a ring is called quasi-Armendariz if whenever and in satisfy , we have for every and ) and provide rich classes of non-semiprime quasi-Armendariz rings. Let be a unitary ring with an endomorphism and be the free monoid generated by with added, and be a factor of setting certain monomial in to , enough so that, for some natural number , . In this paper, we give a sufficient condition for a ring such that the skew m
We say that a -ring R is a generalized quasi-Baer -ring if for any ideal I of the right annihilator of is generated, as a right ideal, by a projection, for some positive integer n depending on I. A unital -ring R is left primary if and only if R is a generalized quasi-Baer -ring with no nontrivial central projections. We study basic properties of such rings and we prove their permanence properties such as the Morita invariance. We show that this notion is well-behaved with respect to polynomial extensions and certain triangular matrix extensions and group rings. A sheaf representation for such -rings is also proved. We obtain algebraic examples which are generalized quasi-Baer -rings but are not quasi-Baer -rings. We show that for pre-C*-
A ring R is called weakly principally quasi Baer or simply (weakly pq-Baer) if the right annihilator of a principal right ideal is right s-unital by right semicentral idempotents, which implies that R modulo the right annihilator of any principal right ideal is flat. We study the relationship between the weakly pq-Baer property of a ring R and those of the skew inverse series rings R ((x− 1; σ, δ)) and R [[x− 1; σ, δ]], for any automorphism σ and derivation δ of R. Examples to illustrate and delimit the theory are provided.
The aim of this article is to determine entirely the Jordan automorphisms of generalized matrix rings of Morita contexts. Necessary and sufficient conditions are obtained for an -linear map on a general Morita context to be a Jordan homomorphism. Moreover, some conditions are studied, under which, any Jordan automorphism of a general Morita context is either an automorphism or an anti-automorphism.
It is well known that when a ring R satisfies ACC on right annihilators of elements, then the right singular ideal of R is nil, in this case, we say R is right nil-singular. Many classes of rings whose singular ideals are nil, but do not satisfy the ACC on right annihilators, are presented and the behavior of them is investigated with respect to various constructions, in particular skew polynomial rings and triangular matrix rings. The class of right nil-singular rings contains π-regular rings and is closed under direct sums. Examples are provided to explain and delimit our results.
Let be an associative ring with identity. A right -module is said to have Property (), if each finitely generated ideal has a nonzero annihilator in . Evans [Zero divisors in Noetherian-like rings, Trans. Amer. Math. Soc. 155(2) (1971) 505–512.] proved that, over a commutative ring, zero-divisor modules have Property (). We study and construct various classes of modules with Property (). Following Anderson and Chun [McCoy modules and related modules over commutative rings, Comm. Algebra 45(6) (2017) 2593–2601.], we introduce -dual McCoy modules and show that, for every strictly totally ordered monoid , faithful symmetric modules are -dual McCoy. We then use this notion to give a characterization for modules with Property (). For a fa
A ring R is called left AIP if R modulo the left annihilator of any ideal is flat. In this paper, we characterize a module MR for which the endomorphism ring is left AIP. We say a module MR is endo-AIP (resp. endo-APP) if M has the property that “the left annihilator in of every fully invariant submodule of M (resp. for every ) is pure as a left ideal in ”. The notion of endo-AIP (resp. endo-APP) modules generalizes the notion of Rickart and p.q.-Baer modules to a much larger class of modules. It is shown that every direct summand of an endo-AIP (resp.endo-APP) module inherits the property and that every projective module over a left AIP (resp. APP)-ring is an endo-AIP (resp. endo-APP) module.
We call a ring generalized right -Baer, if for any projection invariant left ideal of , the right annihilator of is generated, as a right ideal, by an idempotent, for some positive integer , depending on . In this paper, we investigate connections between the\g -Baer rings and related classes of rings (eg, -Baer, generalized Baer, generalized quasi-Baer, etc.) In fact, generalized right -Baer rings are special cases of generalized right quasi-Baer rings and also are a generalization of -Baer and generalized right Baer rings. The behavior of the generalized right -Baer condition is investigated with respect to various constructions and extensions. For example, the trivial extension of the generalized right -Baer ring and the full matrix r
A module M is said to be generalized extending if for every submodule there exists a direct summand D of M containing N such that D/N is a singular module. In this note we prove that a ring R is right self-injective if and only if the triangular ring is right generalized extending. This answers a question which was raised in A. Akalan, G.F. Birkenmeier, A. Tercan, Characterizations of extending modules and -extending generalized triangular matrix rings, Commun. Algebra 40 (2012), 1069–1085.
In this paper, we provide new examples of Banach∗-subalgebras of the matrix algebra M n (A). For any involutive algebra, we define two involutions on the triangular matrix extensions. We prove that the triangular matrix algebras over any commutative unital C∗-algebra, are Banach∗-algebras and that the primitive ideals of these algebras and some of their Banach∗-subalgebras are all maximal.
Abstract: We call a ring R generalized right π-Baer, if for any projection invariant left ideal Y of R, the right annihilator of Y n is generated, as a right ideal, by an idempotent, for some positive integer n, depending on Y. In this paper, we investigate connections between the generalized π-Baer rings and related classes of rings (eg, π-Baer, generalized Baer, generalized quasi-Baer, etc.) In fact, generalized right π-Baer rings are special cases of generalized right quasi-Baer rings and also are a generalization of π-Baer and generalized right Baer rings. The behavior of the generalized right π-Baer condition is investigated with respect to various constructions and extensions. For example, the trivial extension of a generalized
A ring R with an endomorphism σ is called σ-skew McCoy, if for any zero-divisor f(x) in the skew polynomial ring R[x; σ], there exists a nonzero element with f(x)c = 0. In this note, we show that there exists a ring R and an endomorphism σ such that the matrix ring M2(R) is σ-skew McCoy. This gives a negative answer to the question posed in “A. R. Nasr-Isfahani, On semiprime right Goldie McCoy rings, Commun. Algebra 42 (2014) 1565-1570”.
Let SMR be an (S, R)-bimodule of the rings R and S. We determine the associated primes of a formal triangular matrix ring T=(R 0 MS). Indeed, we show that A ss (TT)={(A ss ((R⊕ M) R) 0 MS)}∪{(R 0 MA ss (l S (M)))}. We then obtain necessary and sufficient conditions for the tertiary decomposition theory to exist on a module over an arbitrary ring. Consequently, we classify all the tertiary right ideals of the formal triangular matrix rings.
Let R be a ring,() a strictly ordered monoid, and a monoid homomorphism. In [18], Mazurek, and Ziembowski investigated when the skew generalized power series ring is a domain satisfying the ascending chain condition on principal left (resp. right) ideals. Following [18], we obtain necessary and sufficient conditions on R, S and such that the skew generalized power series ring is a right or left Archimedean domain. As particular cases of our general results we obtain new theorems on the ring of arithmetical functions and the ring of generalized power series. Our results extend and unify many existing results.
In this paper it is shown that, for a module M over a ring R with S= E n d R (M), the endomorphism ring of the R [x]-module M [x] is isomorphic to a subring of S [[x]]. Also the endomorphism ring of the R [[x]]-module M [[x]] is isomorphic to S [[x]]. As a consequence, we show that for a module M R and an arbitrary nonempty set of not necessarily commuting indeterminates X, M R is quasi-Baer if and only if M [X] R [X] is quasi-Baer if and only if M [[X]] R [[X]] is quasi-Baer if and only if M [x] R [x] is quasi-Baer if and only if M [[x]] R [[x]] is quasi-Baer. Moreover, A module M R with IFP, is Baer if and only if M [x] R [x] is Baer if and only if M [[x]] R [[x]] is Baer. It is also shown that, when M R is a finitely generated module, an