By Pat Goeters (Editor), Overtoun M.G. Jenda (Editor)

ISBN-10: 142001076X

ISBN-13: 9781420010763

ISBN-10: 1584885521

ISBN-13: 9781584885528

In honor of Edgar Enochs and his venerable contributions to a huge variety of subject matters in Algebra, best researchers from around the globe amassed at Auburn collage to document on their most modern paintings and trade principles on a few of brand new most efficient examine themes. This rigorously edited quantity offers the refereed papers of the individuals of those talks in addition to contributions from different veteran researchers who have been not able to attend.

These papers replicate a few of the present themes in Abelian teams, Commutative Algebra, Commutative earrings, workforce concept, Homological Algebra, Lie Algebras, and Module conception. available even to starting mathematicians, lots of those articles recommend difficulties and courses for destiny examine. This quantity is an exceptional addition to the literature and a worthy guide for starting in addition to pro researchers in Algebra.

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**Extra resources for Abelian Groups, Rings, Modules, and Homological Algebra**

**Sample text**

If A is slender, then direct summands of A I are A-reflexive as long as I has non-measurable cardinality. In general, A-reflexive modules have a zero A-radical. An exact sequence 0 → B → C → M → 0 of right R-modules is A-cobalanced if the induced sequence 0 → M ∗ → C ∗ → B ∗ → 0 of left E-modules is exact. 1 ([4]) Let A be a slender R-module of non-measurable cardinality. The following are equivalent for a right R-module M of non-measurable cardinality: a) M is A-reflexive. b) There exists an A-cobalanced sequence 0 → M → A I → N → 0 with R A (N ) = 0 and |I | non-measurable.

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 16 19 20 20 Abstract In this paper, we study an invariant (R) introduced by Scott Chapman to measure how far an HFD R is from being a UFD. We show that if either R contains a prime element or R is a Krull domain with finite divisor class group, then R is a UFD if and only if (R) = 0. However, we give an example of an atomic integral domain R with (R) = 0 which is not an HFD.

Proof We show that μ(R S , n) ⊆ μ(R, n) for each positive integer n. Thus (R S , n) ≤ (R, n) for each positive integer n, and hence (R S ) ≤ (R). Let m ∈ μ(R S , n). Then m = η RS (x) for some x ∈ R S with l RS (x) = n. Write x = ux , where x ∈ R, u ∈ U (R S ), and (x , t ) = 1 for all t ∈ S. 4]). Thus m ∈ μ(R, n). Suppose that there is some prime p ∈ R \ S. Let m ∈ μ(R, n). Then m = η R (x) for some x ∈ R with l R (x) = n. Write x = sx , where s ∈ S and (x , t ) = 1 for all t ∈ S. Let l R (s) = k, and set z = pk x .

### Abelian Groups, Rings, Modules, and Homological Algebra by Pat Goeters (Editor), Overtoun M.G. Jenda (Editor)

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