Let R be a ring, M be a left R-module and Spec(RM) be the collection of all prime submodules of M. In this paper and its sequel, we in- troduce and study a generalization of the Zariski topology of rings to modules and call it classical Zariski topology of M. Then we investigate the interplay between the module-theoretic properties of M and the topological properties of Spec(RM). Modules whose classical Zariski topology is respectively T1, Hausdor or cofinite are studied, and several characterizations of such mod- ules are given. We investigate this topological space from the point of view of spectral spaces (that is, topological spaces homeomorphic to the prime spec- trum of a commutative ring equipped with the Zariski topology). We show that Spec(RM) is always a T0-space and each finite irreducible closed subset of Spec(RM) has a generic point. Then by applying Hochster's characterization of a spectral space, we show that for each left R-module M with finite spec- trum, Spec(RM) is a spectral space. In Part II we shall continue the study of this construction.

/pdf/classical-zariski-topology-of-modules-and-spectral-spaces-i-3as7fl0z0i.pdf

Classical zariski topology of modules and spectral spaces i

A generalization of the classical Krull dimension for modules

We carry out a study of rings R for which HomR (M;N) 6= 0 for all nonzero N ≤ MR. Such rings are called retractable. For a retractable ring, Artinian condition and having Krull dimension are equivalent. Furthermore, a right Artinian ring in which prime ideals commute is precisely a right Noetherian retractable ring. Retractable rings are characterized in several ways. They form a class of rings that properly lies between the class of pseudo-Frobenius rings, and the class of max divisible rings for which the converse of Schur's lemma holds. For several types of rings, including commutative rings, retractability is equivalent to semi-Artinian condition. We show that a Kothe ring R is an Artinian principal ideal ring if and only if it is a certain retractable ring, and determine when R is retractable.

/pdf/on-rings-whose-modules-have-nonzero-homomorphisms-to-nonzero-fk7dn979k6.pdf

On rings whose modules have nonzero homomorphisms to nonzero submodules

A right module M over a ring R is said to be retractable if Hom R (M, N) ≠ 0 for each nonzero submodule N of M. We show that M ⊗ R RG is a retractable RG-module if and only if M R is retractable for every finite group G. The ring R is (finitely) mod-retractable if every (finitely generated) right R-module is retractable. Some comparisons between max rings, semiartinian rings, perfect rings, noetherian rings, nonsingular rings, and mod-retractable rings are investigated. In particular, we prove ring-theoretical criteria of right mod-retractability for classes of all commutative, left perfect, and right noetherian rings.

Mod-Retractable Rings

Abstract For a fixed natural number an element a of a ring R is m-nil clean if where and b is nilpotent; if further eb = be, a is called strongly m-nil clean. The ring R is called m-nil clean (resp., strongly m-nil clean) if each of its elements is m-nil clean (resp., strongly m-nil clean). For a fixed natural number the strongly m-nil clean rings are a big subclass of the periodic rings which contains the class of strongly nil clean rings. The strongly m-nil cleanness of the tensor product of algebras, matrix rings, Morita contexts, and group rings is discussed in detail. Our results extend several existing results. Examples are provided to illustrate our results.

Rings whose elements are sums of m-potents and nilpotents

All rings are assumed to be associative and (except for nil-rings and some stipulated cases) to have nonzero identity elements. Expressions such as a “Noetherian ring” mean that the corresponding right and left conditions hold. A module is said to be simple if it does not have nonzero proper submodules. A submodule N of a module M is called a maximal submodule (in M) if the factor module M/N is simple. A ring A is called a right max ring (or a right Bass ring) if every nonzero right A-module has a maximal submodule. A ring is said to be orthogonally finite if it does not contain an infinite set of nonzero orthogonal idempotents. Bass [11] proved that a ring with the minimum condition on principal left ideals is an orthogonally finite right max ring. Cozzens [25] and Koifman [61] constructed examples of orthogonally finite right max rings which do not satisfy the minimum condition on principal left ideals. This review contains some new and some old results related to max rings. We present the necessary notation and definitions. Let A be a ring. We denote the center of the ring A by C(A). A subset B of A is said to be central (in A) if B ⊆ C(A). If M and N are two modules, then Hom(M,N) denotes the Abelian group formed by all homomorphisms M → N . For a module M , we denote by End(M) the endomorphism ring Hom(M,M) of M . For a subset B of A, the right annihilator and the left annihilator of B in A are denoted by rA(B) and A(B), respectively. We can omit the subscripts if the situation is obvious. An element a of A is said to be right regular (resp. left regular) in A if r(a) = 0 (resp. (a) = 0). A ring A is said to be regular if for every element a of A, there exists an element b of A such that a = aba. A ring A is said to be right weakly regular if B2 = B for every right ideal B of A. A ring without nonzero nilpotent ideals is called a semiprime ring. A ring is called a prime ring if the product of any two of its nonzero ideals is not equal to zero. A ring A is said to be simple if each of its nonzero ideals coincides with A. A ring is called a domain if the product of any two of its nonzero elements is not equal to zero. For a ring A, an ideal P of A is called a prime (resp. semiprime and completely prime) ideal if the factor ring A/P is a prime ring (resp. a semiprime ring and a domain). A ring A is called a P.I.-ring (or a ring with polynomial identity) if A satisfies the polynomial identity f(x1, . . . , xn) = 0, where f(x1, . . . , xn) is a polynomial in noncommutative variables with coefficients in the ring of integers Z, and Z coincides with its ideal generated by the coefficients of f(x1, . . . , xn). A submodule N of a module M is said to be essential (in M) if N has nonzero intersection with any nonzero submodule of the module M . In this case, we say that M is an essential extension of the module N . A submodule N of a module M is called a superfluous submodule (in M) if N +M ′ = M for every proper submodule M ′ of M .

Rings Whose Nonzero Modules Have Maximal Submodules

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Rings Whose Nonzero Modules Have Maximal Submodules