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Let R be a commutative ring with non-zero identity and M be a unitary R-module. The goal of this paper is to extend the concept of 1-absorbing primary ideals to 1-absorbing primary submodules. A proper submodule N of M is said to be a 1-absorbing primary submodule if whenever non-unit elements a, b ∈ R and m ∈ M with abm ∈ N, then either ab ∈ (N :RM) or m ∈ M − rad(N). Various properties and chacterizations of this class of submodules are considered. Moreover, 1-absorbing primary avoidance theorem is proved.

A gamma ring(Γ− ring) is ΓR-module and W is proper ΓR-submodule. Let W be a proper ΓR-submodule of an ΓR-module is called nearly prime, for short N-P.ΓR-submodule if whenever and rγ m ∈ W implies that either or where is the Jacobson radical of. The goal of this work is to offer an intriguing way of defining and studying nearly P.ΓR-submodule of ΓR-module, for short N-P.ΓR-submodule of ΓR-module result N-P.ΓR-submodule of ΓR-module which represent generalization N-prime submodules. [ABSTRACT FROM AUTHOR]

We study direct-sum decompositions of torsion-free, finitely generated modules over a (commutative) Bass ring R through the factorization theory of the corresponding monoid T(R). Results of Levy–Wiegand and Levy–Odenthal together with a study of the local case yield an explicit description of T(R). The monoid is typically neither factorial nor cancellative. Nevertheless, we construct a transfer homomorphism to a monoid of graph agglomerations—a natural class of monoids serving as combinatorial models for the factorization theory of T(R). As a consequence, the monoid T(R) is transfer Krull of finite type and several finiteness results on arithmetical invariants apply. We also establish results on the elasticity of T(R) and characterize when T(R) is half-factorial. (Factoriality, that is, torsion-free Krull–Remak–Schmidt–Azumaya, is characterized by a theorem of Levy–Odenthal.) The monoids of graph agglomerations introduced here are also of independent interest. [ABSTRACT FROM AUTHOR]

Modules in which every submodule is isomorphic to a direct summand is called virtually semisimple. In this article, we carry out a study of virtually semisimple modules over a commutative ring R. A structure theorem of finitely generated virtually semisimple R-modules is given. Also, it is proven that if every submodule of an R-module M is virtually semisimple, then M = ⊕ i ∈ I M i where I is an index set and for each i, MinAss (M i) = { P i } , R / P i is a principal ideal domain and M i = S i ⊕ T i with semisimple and torsionfree (R / P i) -modules Si and Ti, respectively. As an application of our "structure theorem," we give a characterization of commutative rings for which each proper ideal is virtually semisimple. [ABSTRACT FROM AUTHOR]

The spline complex whose top homology is the algebra of mixed splines over a fan was introduced by Schenck–Stillman as a variant of a complex of Billera. In this paper we analyze the associated primes of homology modules of this complex. In particular, we show that all such primes are linear. We give two applications to computations of dimensions. The first is a computation of the third coefficient of the Hilbert polynomial of the algebra of mixed splines over a fan, including cases where vanishing is imposed along arbitrary codimension one faces of the boundary of the fan, extending computations by Geramita–Schenck in the simplicial case and McDonald–Schenck in the polytopal case. The second is a description of the fourth coefficient of the Hilbert polynomial of the algebra of mixed splines over simplicial fans. We use this to re-derive a result of Alfeld–Schumaker–Whiteley on the generic dimension of C 1 tetrahedral splines in large degree and indicate via an example how this description may be used to give the fourth coefficient in particular non-generic configurations. [ABSTRACT FROM AUTHOR]

The toric fiber product is an operation that combines two ideals that are homogeneous with respect to a grading by an affine monoid. The Segre product is a related construction that combines two multigraded rings. The quotient ring by a toric fiber product of two ideals is a subring of the Segre product, but in general this inclusion is strict. We contrast the two constructions and show that any Segre product can be presented as a toric fiber product without changing the involved quotient rings. This allows to apply previous results about toric fiber products to the study of Segre products. We give criteria for the Segre product of two affine toric varieties to be dense in their toric fiber product, and for the map from the Segre product to the toric fiber product to be finite. We give an example that shows that the quotient ring of a toric fiber product of normal ideals need not be normal. In rings with Veronese type gradings, we find examples of toric fiber products that are always Segre products, and we show that iterated toric fiber products of Veronese ideals over Veronese rings are normal. [ABSTRACT FROM AUTHOR]

A well-known result of Köthe and Cohen-Kaplansky states that a commutative ring R has the property that every R-module is a direct sum of cyclic modules if and only if R is an Artinian principal ideal ring. This motivated us to study commutative rings for which every ideal is a direct sum of cyclic modules. Recently, in Behboodi et al. Commutative Noetherian local rings whose ideals are direct sums of cyclic modules (J. Algebra 345:257-265, ) the authors considered this question in the context of finite direct products of commutative Noetherian local rings. In this paper, we continue their study by dropping the Noetherian condition. [ABSTRACT FROM AUTHOR]

We study the symbolic blowup ring of a prime ideal defining a monomial curve in the power series ring in 3 variables over a field. We characterize the conditions required to have the symbolic blowup generated in degree 4 when the monomial curve is self-linked. [ABSTRACT FROM AUTHOR]

It is proved that a commutative ring is clean if and only if it is Gelfand with a totally disconnected maximal spectrum. It is shown that each indecomposable module over a commutative ring R satisfies a finite condition if and only if RP is an Artinian valuation ring for each maximal prime ideal P. Commutative rings for which each indecomposable module has a local endomorphism ring are studied. These rings are clean and elementary divisor rings. It is shown that each commutative ring R with a Hausdorff and totally disconnected maximal spectrum is local-global. Moreover, if R is arithmetic, then R is an elementary divisor ring. [ABSTRACT FROM AUTHOR]

Let M be a finitely generated module of rank r and finite projective dimension over a commutative Noetherian local ring S . We show that M has a free direct summand of rank r if and only if its Buchsbaum–Eisenbud invariant J ( M ) is a principal ideal. As an application, we consider the problem of finding rigid triples. Recall that the module M is said to be rigid if for each i = 0 and each finitely generated S -module N the condition implies for every j = i . A triple of positive integers b = ( b 0 , b 1 , b 2 ) is said to be rigid if for every commutative Noetherian local ring S one has that every S -module with minimal free resolution of the form 0 ? S b 2 ? S b 1 ? S b 0 ? 0 is rigid. We use our criterion for free direct summands of maximal rank to derive a necessary and sufficient condition for a triple b to be rigid in terms of the rigidity of the universal modules of type b over complete regular local rings. In particular, we exhibit the rigidity of any triple of the form ( k , m + 1, m ), thus providing a new class of rigid modules. [ABSTRACT FROM AUTHOR]

Let R be an integral domain, and let A be an "almost" finitely presented R-moduk such that R/Ann (A) is a finite direct sum of quasilocal rings. We characterize the relationship between the modules that can occur as kernels of finite rank projective presentations of A, and we give a criterion for when local isomorphism of torsionless modules implies an equivalence of the two modules up to summands of projective R-modules. [ABSTRACT FROM AUTHOR]

Let K be a ring with an identity 1 ≠ 0 and M, L two unitary K-modules. Then, for any additive mapping f:M →L, the set H:={ α ∈ K ∣ f(αx)=αf(x) for all x ∈ M} forms a subring of K, the 'homogeneity ring' of f. It is shown that, for M ≠ {0}, L ≠ {0} and any subring S of K for which M is a free S-module, there exists an additive mapping f:M→L such that H =S. This result is applied to the four Cauchy functional equations, and it leads also to an answer to the question as to whether it is possible to introduce on M a multiplication ·: M × M → M making M into a ring but not into a K-algebra. [ABSTRACT FROM AUTHOR]