Semiregular Rings Research Articles

Rings characterized by projectivity classes

Projectivity classes, which dualize injectivity classes (cf.[ll]), are introduced and some examples are given. Characterizations of hereditary rings, semisimple rings, Noetherian rings, (semi-)perfect rings, quasi-perfect rings, semiregular rings, semiregular modules and F- semiperfect modules using projectivity classes are given. Finally, for a projectivity class P, P-projective covers are defined and similar results with “quasi-projective cover” substituted by “T-projective cover” will still hold. Our results unify and generalize several well known results by Golan, Huynh and Smith, Rangaswamy and Vanaja, Tiwary and Pandeya, and Xue.

Commutative semi-coherent and semi-regular rings

Throughout this paper all rings considered will be commutative rings with identity. We shall consider the consequences for a ring of the flatness of certain classes of its modules. Since flatness is determined by the finitely generated ideals of a ring we shall in a real sense be studying conditions on these ideals. A coherent ring is a ring whose finitely generated ideals are finitely presented. Chase has proved [3, Theorem 2.11 that a ring is coherent if and only if any direct product of its flat modules is flat. In [8, Theorem 1 J we proved that a ring R is coherent if and only if Hom,(B, C) is flat whenever B and C are injective R-modules. We define a ring R to be semicoherent if Hom,(B, C) is a submodule of a flat R-module whenever B and C are injective R-modules. This is a property shared by both integral domains and Noetherian rings. We prove in Proposition 1.1 that a reduced ring R is semi-coherent if and only if min R is compact. Proposition 1.3 states that if R is a semi-coherent ring, then a finitely generated submodule of a free R-module is flat if and only if it is projective. A regular ring is a ring whose finitely generated ideals are direct summands of the ring. Equivalently it is a ring such that every one of its modules is flat. Inspired by our definition of semi-coherent we define a ring to be semi-regular if every module is a submodule of a flat module. Proposition 2.2 states that if the total ring of quotients of a ring R is semiregular, then R is a semi-coherent ring. An even stronger connection with the notion of coherence is provided by Proposition 2.3: A ring R is semiregular if and only if R is coherent and RM is semi-regular for every maximal ideal M of R. Proposition 2.7 states that if R is a reduced ring, then R is semi-regular if and only if R is regular; and Proposition 3.4 states that if R is a Noetherian

Semilocal and semiregular group rings

Semilocal and semiregular group rings

Ideals defined by matrices and a certain complex associated with them

For each matrix, whose elements belong to a commutative ring with an identity element, there is defined a free complex. This complex is a generalization of the standard Koszul complex, which corresponds to the case of a matrix with only a single row. The applications are to certain ideals defined by the maximal subdeterminants of a matrix. It is found that such an ideal has finite projective dimension whenever its grade reaches a certain greatest value (depending on the dimensions of the matrix) and that, in these circum stances, the complex provides a free resolution of the correct length. For semi-regular ( = M acaulayCohen) rings this leads to a theorem on unmixed ideals. In the case of arbitrary Noetherian rings, a general theorem on rank is proved.

SEMI-REGULAR RINGS AND SEMI-REGULAR IDEALS

SEMI-REGULAR RINGS AND SEMI-REGULAR IDEALS Get access D. G. NORTHCOTT D. G. NORTHCOTT Sheffield Search for other works by this author on: Oxford Academic Google Scholar The Quarterly Journal of Mathematics, Volume 11, Issue 1, 1960, Pages 81–104, https://doi.org/10.1093/qmath/11.1.81 Published: 01 January 1960 Article history Received: 10 December 1958 Published: 01 January 1960

Semi‐regular local rings

In the following pages there will be found an account of the properties of a certain class of local rings which are here termed semi-regular local rings . As this name will suggest, these rings share many properties in common with the more familiar regular local rings, but they form a larger class and the characteristic properties are preserved under a greater variety of transformations. The first occasion on which these rings were studied by the author was in connection with a problem concerning the irreducibility of certain ideals, but about the same time they were investigated in much greater detail by Rees [7] and in quite a different connection. In his discussion, Rees made considerable use of the ideas and techniques of homological algebra. Here a number of the same results, as well as some additional ones, are established by quite different methods. The essential tools used on this occasion are the results obtained by Lech [3] in his important researches concerning the associativity formula for multiplicities. Before describing these, we shall first introduce some notation which will be used consistently throughout the rest of the paper.