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The well-known notion of an extending module is closely linked to that of a Baer module. A right R -module M is called extending if every submodule of M is essential in a direct summand. On the other hand, a right R -module M is called Baer if for all N ≤ M , l S (N) ≤ ⊕ S S where S = End R (M). In 2004, Rizvi and Roman generalized a result of [A. W. Chatters and S. M. Khuri, Endomorphism rings of modules over nonsingular CS rings, J. London Math. Soc.  21(2) (1980) 434–444.] in terms of modules and showed the connections between Baer and extending modules via the result: "a module M is -nonsingular extending if and only if M is -cononsingular Baer". M R is called -nonsingular if ∀ φ ∈ S such that Ker φ ≤ e M , φ = 0. Moreover, M R is called -cononsingular if for any N ≤ M with φ N ≠ 0 for all 0 ≠ φ ∈ S , implies N ≤ e M. In view of this result, every Baer module which happens to be -cononsingular will automatically become an extending module. In this paper, our main focus is the study of -cononsingularity of modules. Our investigations are also motivated by the fact that very little is known about the notion of -cononsingularity while sufficient knowledge exists about the other three remaining notions in the preceding result. Moreover, we introduce the notion of special extending (or sp-extending, for short) of a module and show that the class of -cononsingular modules properly contains the class of extending modules and the class of special extending modules. Among other results, we obtain a new analogous version for the Rizvi–Roman's result which illustrates the close connections between Baer and extending modules. Examples illustrating the notions and delimiting our results are provided. [ABSTRACT FROM AUTHOR]

This paper defines the Euler–Poincaré characteristic of joint reductions of ideals which concerns the maximal terms in the Hilbert polynomial; characterizes the positivity of mixed multiplicities in terms of minimal joint reductions; proves the additivity and other elementary properties for mixed multiplicities. The results of the paper together with the results of [Thanh and Viet, Mixed multiplicities of maximal degrees, J. Korean Math. Soc.55(3) (2018) 605–622] seem to show a natural and nice picture of mixed multiplicities of maximal degrees. [ABSTRACT FROM AUTHOR]

Let τ be a finite type hereditary torsion theory on the category of all modules over a commutative ring. The purpose of this paper is to give two applications of Nagata rings and modules in the sense of Jara [Nagata rings, Front. Math. China10 (2015) 91–110]. First they are used to obtain Chase's Theorem for τ -coherent rings. In particular, we obtain the w -version of Chase's Theorem, where w is the classical star operation in ideal theory. In the second half, we apply they to characterize τ -flatness in the sense of Van Oystaeyen and Verschoren [Relative Invariants of Rings-The Commutative Theory, Monographs and Textbooks in Pure and Applied Mathematics, Vol. 79 (Marcel Dekker, Inc., New York, 1983)]. [ABSTRACT FROM AUTHOR]

In this paper, we characterize quasi-Yetter–Drinfeld modules over a Hopf algebra H , which was introduced in [Y. Bazlov and A. Berenstein, Braided doubles and rational Cherednik algebras, Adv. Math. 220 (2009), 1466–1530]. We first show that the quasi-Drinfeld center of the category of H -modules is equivalent to the category H H 𝒬 𝒴 𝒟 of quasi-Yetter–Drinfeld modules. Next, we prove that H H 𝒬 𝒴 𝒟 is equivalent to the category of generalized Hopf bimodules. Finally, we show that H H 𝒬 𝒴 𝒟 is also equivalent to the category of quasi-coactions over some Majid's braided group if H is quasi-triangular. [ABSTRACT FROM AUTHOR]

In this paper, we give a recursive formula for the interesting PBW basis E A of composition subalgebras ℭ ▵ (n) of Ringel–Hall algebras ℌ ▵ (n) of cyclic quivers after [Generic extensions and canonical bases for cyclic quivers, Canad. J. Math. 59(6) (2007) 1260–1283], and another construction of canonical bases of U v + ( 𝔰 𝔩 ̂ n) from the monomial bases m (A) following [Multiplication formulas and canonical basis for quantum affine, 𝔤 𝔩 n , Canad. J. Math. 70(4) (2018) 773–803]. As an application, we will determine all the canonical basis elements of U v + ( 𝔰 𝔩 ̂ 2) associated with modules of Loewy length ≤ 3. Finally, we will discuss the canonical bases between Ringel–Hall algebras and affine quantum Schur algebras. [ABSTRACT FROM AUTHOR]

Let R ⊂ A be factorial domains containing ℚ. In this paper, we give a criterion, in terms of locally nilpotent derivations, for A to be R -isomorphic to R [ v , w ] / (c w − h (v)) , where c ∈ R is nonzero and h (v) ∈ R [ v ]. As a consequence, we retrieve a recent result due to Masuda [Families of hypersurfaces with noncancellation property, Proc. Amer. Math. Soc.145(4) (2017) 1439–1452] characterizing Danielewski hypersurfaces whose coordinate ring is factorial. We also apply our criterion to the study of triangularizable locally nilpotent R -derivations of the polynomial ring in two variables over R. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to establish a real Nullstellensatz, a Positivstellensatz and a Nichtnegativstellensatz for matrices over a commutative ring. The Stellensätze in this paper may be regarded as certain generalizations of the abstract Stellensätze for commutative rings. [ABSTRACT FROM AUTHOR]

The modified quantum algebra associated to a quantum algebra was introduced by Lusztig. has a remarkable basis, which was defined by Lusztig, called the canonical basis. In this paper, we give an explicit description of all elements of the canonical basis of for type . [ABSTRACT FROM AUTHOR]

The concept of internally cancellable rings has been extensively studied in the literature. This paper seeks to continue the study of these rings and find some new characterizations. It is proved that is 'IC', if and only if for each regular element , and idempotent element with , there exists such that is a unit (alternatively, unit-regular element) in and . In case the ring has the summand sum property, we indicate that is IC, if and only if for each regular element , and element with , there exists an idempotent , such that is a unit in and . [ABSTRACT FROM AUTHOR]

In this paper, we generalize the main results of Guo and Isaacs, Archiv der Mathematik 105 (2015) 215-222. We also obtain some criteria for -nilpotency of a finite group. [ABSTRACT FROM AUTHOR]

Let be an integral domain with quotient field . Call an almost GCD domain (AGCD domain) if for all , there is an integer such that is principal. We say that is a locally AGCD domain (resp., strongly locally AGCD domain) if is an AGCD domain for all maximal ideals of (resp., for all , there is an integer such that is locally principal). In this paper, we study some ring-theoretic properties of locally and strongly locally AGCD domains. Let be an indeterminate over . We use the ring for an integer to give some examples of locally and strongly locally AGCD domains. [ABSTRACT FROM AUTHOR]

In this paper, we prove an Artin type theorem for semigroups. Namely, we consider the concepts of hyperalternative and hyperassociative semigroups and prove that every two elements in any hyperalternative semigroup generate a hyperassociative subsemigroup. As a consequence, we characterize all hyperalternative semigroups, and prove that these semigroups form a variety of semigroups with four identities. [ABSTRACT FROM AUTHOR]

In [H. P. Tong-Viet, Simple classical groups of Lie type are determined by their character degrees, J. Algebra 357 (2012) 61-68] the following question arose: Question. Which groups can be uniquely determined by the structure of their complex group algebras? It is proved that every quasisimple group except covers of the alternating groups is uniquely determined up to isomorphism by the structure of , the complex group algebra of . One of the next natural groups to be considered are the characteristically simple groups. In this paper, as the first step in this investigation we prove that if is an odd prime number, then is uniquely determined by the structure of its complex group algebra. [ABSTRACT FROM AUTHOR]