Nonzero Divisors Research Articles

Localization for flat modules in algebraic K-theory

There are two localization theorems in algebraic K-theory which yield long esact sequences for the K-groups of certain triples of exact categories: One handles the quotient of an Ahelian category [9], and the other relates a ring and its localization by nonzero divisors. We present here a localization theorem for certain categories of R-modules flat over a base ring d (Theorem 4.8). The problem encountered in [6] was the lack of suitable non-Abelian quotient categories for the filtration by relative codimension of support: We find a suitable quotient category and use techniques of Quillen to obtain the long exact sequence. The proof is longer than the proof in [9] of the localization theorem for Abelian catcgorics, and dots not retain the symmetry between monies and epis. This asymmetry leads to the introduction of relatively torsion-free modules, for they tend to map into or onto enough other modules. Useful tools include a generalization of Waldhauscn’s cofinality theorem and a modification of Quillen’s resolution theorem. The quotient category obtained is not very computable; a slightly larger exact category is more so. The K-groups are the same if cofinality can be established, but we can only do this for codimension <l (Theorem 5.2). This is good enough to show that the first two sheaves in the resolution of .X,(X) constructed in [6j are flasque (Proposition 6.4). If X is smooth over S of relative dimension 2, and S is finite, then we can see that A”(X) :: H’(X, Z,) is g enerated by classes of coherent C’,-modules finite and free over CS , and the relations can be described (Section 9) with no reference to higher K-theory. Recall that when 5’ is a field, -if’(X) is the group of zero cycles on X modulo rational equivalence. When S is not a field, A’(S) may yield information about the infinitesimal str-ucture of the cycle class group, as in [2]; moreover A”(X) is part of a graded ring A’(X) which might harbor Chern classes for vector bundles on the singular variety X. The relations for A2(X) can be found by computing boundary maps from higher K-theory and using the fact that K, of a local ring is generated by symbols. They are those given by exact sequences and those of the form c(y, f, g) q -= 0, 463

CPI-Extensions: Overrings of Integral Domains with Special Prime Spectrums

Throughout this paper the term ring will denote a commutative ring with unity and the term integral domain will denote a ring having no nonzero divisors of zero. The set of all prime ideals of a ring R can be viewed as a topological space, called the prime spectrum of R, and abbreviated Spec (R), where the topology used is the Zariski topology [1, Definition 4, § 4.3, p. 99]. The set of all prime ideals of R can also be viewed simply as aposet - that is, a partially ordered set - with respect to set inclusion. We will use the phrase the pospec of R, or just Pospec (/v), to refer to this partially ordered set.

Classical quotient rings

Throughout R is a ring with right singular ideal Z ( R ) Z(R) . A right ideal K of R is rationally closed if x − 1 K = { y ∈ R : x y ∈ K } {x^{ - 1}}K = \{ y \in R:xy \in K\} is not a dense right ideal for all x ∈ R − K x \in R - K . A ring R is a Cl-ring if R is (Goldie) right finite dimensional, R / Z ( R ) R/Z(R) is semiprime, Z ( R ) Z(R) is rationally closed, and Z ( R ) Z(R) contains no closed uniform right ideals. These rings include the quasi-Frobenius rings as well as the semiprime Goldie rings. The commutative Cl-rings have Cl-classical quotient rings. The injective ones are congenerator rings. In what follows, R is a Cl-ring. A dense right ideal of R contains a right nonzero divisor. If R satisfies the minimum condition on rationally closed right ideals then R has a classical Artinian quotient ring. The complete right quotient ring Q (also called the Johnson-Utumi maximal quotient ring) of R is a Cl-ring. If R has the additional property that bR is dense whenever b is a right nonzero divisor, then Q is classical. If Q is injective, then Q is classical.

Annihilators of modules with a finite free resolution

Let A be a commutative ring and let E be an A-module with a finite free resolution (see below for definition). Extending results known previously for noetherian rings, it is shown that ann ( E ) = annihilator {\text {ann}}(E)=\text {annihilator} of E is trivial if and only if the Euler characteristic of E = χ ( E ) &gt; 0 E = \chi (E) &gt; 0 ; in addition, if χ ( E ) = 0 , ann ( E ) \chi (E) = 0,{\text {ann}}(E) is dense (i.e. ann ( ann ( E ) ) = 0 {\text {ann}}({\text {ann}}(E)) = 0 ). Also, a local ring is constructed with its maximal ideal with a finite free resolution but consisting exclusively of zero-divisors and thus, contrary to the noetherian case, one does not always have a nonzero divisor in ann ( E ) {\text {ann}}(E) if χ ( E ) = 0 \chi (E) = 0 . Finally, if E has a finite resolution by (f.g.) projective modules it turns out that ann ( ann ( E ) ) {\text {ann}}({\text {ann}}(E)) is generated by an idempotent element.

The value-semigroup of a one-dimensional Gorenstein ring

In a conversation about [4], 0. Zariski indicated to the author that there should be a relation between Gorenstein rings and symmetric value-semigroups, possibly allowing a new proof for a result of Herzog on complete intersections. In the following note it is shown that this is the case. We use the following facts about Gorenstein rings: If R is a onedimensional noetherian local ring with maximal ideal m and full ring of quotients Q(R), then the following conditions are equivalent: (a) R is a Gorenstein ring (by definition: m contains a nonzero divisor, which generates an irreducible ideal). (b) Each principal ideal, generated by a nonzero divisor, is irreducible. (c) The length of the R-module m-'/R is 1. (d) For each ideal a of R, which contains a nonzero divisor, (a-')-'=a. Here the inverse of an ideal is taken in Q(R). For easy proofs of these equivalences see Berger [3]. Let R be the integral closure of R in Q(R) and f the conductor from R to W. If W is a finitely generated R-module, then f contains a nonzero divisor. If R is Gorenstein, then the length of the R-module R/f is 2d, where d is the length of R/f. Roquette [5] gives the following simple proof: If

On finitely generated flat modules

Introduction. The aim of this paper is to study conditions which reflect the projectivity of a given finitely generated flat module over a commutative ring. The use of the invariant factors of a module (see below for definition) are very appropriate here: By translating the description by Bourbaki [4] of finitely generated projective modules, one can state that projectivity=flatness+finitely generated invariant factors. Since the invariant factors of a flat module are very peculiar (locally they are either (1) or (0)), the presence of almost any other condition on the module precipitates their finite generation. For instance, consider the following two statements. Let M be a finitely generated flat moGule over the commutative ring R: (i) Let S be a multiplicative set in R consisting of nonzero divisors such that MS (localization of M with respect to S) is Rs-projective; then M is projective. (ii) Let J be the Jacobson radical of R and assume that MIJM is R/J-projective; then M is projective. The first is a result by Endo [8] who uses homological algebra in the proof, while the second can be viewed as a generalization of the well-known fact that over a local ring a finitely generated flat module is free. Next we apply these ideas for a look at finitely generated flat ideals. Even though they are not always projective (see below for an example of a principal flat ideal which is not projective) it can be shown this to be the case when the flat ideal is a finite intersection of primary ideals. The criterion mentioned above says that for a finitely generated flat ideal, projectivity is the same as having finitely generated annihilator. For rings with the weakened form of coherency that finitely generated ideals have finitely generated annihilators, one can even show that any finitely generated flat submodule of a projective module is projective. When used to study the prime ideals of a ring R of weak dimension one we arrive at the fact that a finitely generated prime ideal is either maximal or generated by an idempotent. It is also shown that if every principal ideal is projective then R is semihereditary; if moreover every cyclic flat module is projective, then R is a direct sum of finitely many Priufer domains.

Semisimple maximal quotient rings

Notation and Introduction. R denotes an arbitrary associative ring. A right R-module A over R will be denoted AR. BR is a large submodule of AR (AR is an essential extension of BR), if BR is a submodule of AR having nonzero intersection with every nonzero submodule of AR. A right ideal I of R is a large right ideal, if IR is a large submodule of RR. Given AR, Z(AR) is the singular submodule of AR [9], which consists of all those elements of A whose annihilators in R are large right ideals. Following Johnson [9], Q is a right quotient ring of R if Q is a ring with identity containing R as a subring (the identity of Q is the identity of R if R has one) and RR is a large submodule of QR. The quotient rings considered by Goldie in [6], [7] will be called classical quotient rings. Q is a classical right quotient of R if every regular element (nonzero divisor) of R is a unit in Q and every element of Q is of the form ab -1, a, b E R, b regular in R. In general, a ring R need not possess a classical right quotient ring. Goldie [7], has given necessary and sufficient conditions that a ring possess a classical right quotient ring which is semisimple. Here semisimple means semisimple with minimum condition [8]. This paper is concerned with the question of characterizing those rings which have a semisimple maximal right quotient ring [4], [9], [10], [11] and in this case generalizing some simple well-known results about commutative integral domains, their quotient rings and modules over these domains. Johnson [9] has shown that R has a regular maximal right quotient ring Q if and only if Z(RR) = 0, where Q is a regular ring [13] if every finitely generated right (left) ideal of Q is generated by an idempotent. In this case QR is injective [3] as a right R-module, hence the injective hull of R [2]. A ring R has a semisimple maximal right quotient ring Q if and only if Z(RR) =O and dim RR is finite, where a right R-module M is of finite dimension if every direct sum of submodules of M has only finitely many nonzero summands. This is the main result of ?1. In addition another characterization is given for rings which possess a semisimple classical right quotient ring, namely, R has a semisimple classical right quotient ring if and only if Z(RR) = 0 and if I is a large right ideal of R, then there is an element a E I such that aR is a large right ideal of R. If R has a semisimple classical right quotient ring Q, then it is known [3], that