James W. Brewer

Bio: James W. Brewer is an academic researcher from Stellenbosch University. The author has contributed to research in topics: Jordan algebra & Cellular algebra. The author has an hindex of 3, co-authored 3 publications receiving 30 citations.

A note on finitely generated ideals which are locally principal

Abstract: Let R be a commutative ring with identity 1#0 and let A be a nonzero ideal of R. A problem of current interest is to relate the notions of "projective ideal", "flat ideal" and "multiplication ideal". In this note we prove two results which show that the maximal ideals containing the annihilator of A can play an important role in determining the relationship between these concepts. As a consequence we are able to prove that a finitely generated multiplication ideal in a semi-quasi-local ring is principal, that a finitely generated flat ideal having only a finite number of minimal prime divisors is projective and that for Noetherian rings or semihereditary rings, finitely generated multiplication ideals with zero annihilator are invertible. Our notation is essentially that of [4]. In particular, an ideal A of R is said to be a multiplication ideal if whenever B is an ideal of R with BC A, there exists an ideal C of R such that B=AC. There is one deviation from the notation of [4] and that is that we shall denote by A ' the aninihilator of an ideal A. THEOREM 1. Let A be a finitely generated multiplication ideal of R. If A' is contained in onlyfinitely many maximal ideals, then A is principal. PROOF. Let M1, *, Mn be the maximal ideals of R containing A'. For i between 1 and n, if 1l;lji MJA MiA, then MiARMi2 (Tl==1.ji MjA)RM.=ARM.. Since ARM, is finitely generated and since MiRMi is the Jacobson radical of Rm,, it follows from the Nakayama Lemma that ARM.=(O). Thus, ALRM,=(ARM.)'=RM., which contradicts the fact that A'-M. Therefore, for 1 , then we also have that M? (aR: A) since aoAM. Received by the editors March 15, 1971. AMS 1969 subject classifications. Primary 1320; Secondary 1340.

Multiplication Ideals, Multiplication Rings, and the Ring R(X)

Abstract: Let R be a commutative ring with an identity. An ideal A of R is called a multiplication ideal if for every ideal B ⊆ A there exists an ideal C such that B = AC. A ring R is called a multiplication ring if all its ideals are multiplication ideals. A ring R is called an almost multiplication ring if RM is a multiplication ring for every maximal ideal M of R. Multiplication rings and almost multiplication rings have been extensively studied—for example, see [4; 8; 9; 11; 12; 15; and 16].

Class and rank of differential modules

Abstract: A differential module is a module equipped with a square-zero endomorphism. This structure underpins complexes of modules over rings, as well as differential graded modules over graded rings. We establish lower bounds on the class - a substitute for the length of a free complex - and on the rank of a differential module in terms of invariants of its homology. These results specialize to basic theorems in commutative algebra and algebraic topology. One instance is a common generalization of the equicharacteristic case of the New Intersection Theorem of Hochster, Peskine, P. Roberts, and Szpiro, concerning complexes over commutative noetherian rings, and of a theorem of G. Carlsson on differential graded modules over graded polynomial rings.

Ascent of Finite Cohen–Macaulay Type

Abstract: In 1987 F-O Schreyer conjectured that a local ring R has finite Cohen–Macaulay type if and only if the completion R has finite Cohen–Macaulay type We prove the conjecture for excellent Cohen–Macaulay local rings and also show by example that it can fail in general