By Andrei V. Kelarev, R. Gobel, K. M. Rangaswamy, P. Schultz, C. Vinsonhaler

ISBN-10: 0821827510

ISBN-13: 9780821827512

ISBN-10: 1071996193

ISBN-13: 9781071996195

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ISBN-13: 9781741995770

ISBN-10: 3519752352

ISBN-13: 9783519752356

ISBN-10: 5222000516

ISBN-13: 9785222000519

This quantity offers the court cases from the convention on Abelian teams, earrings, and Modules (AGRAM) held on the collage of Western Australia (Perth). integrated are articles according to talks given on the convention, in addition to a number of in particular invited papers. The lawsuits are devoted to Professor Laszlo Fuchs. The booklet encompasses a tribute and a evaluate of his paintings by way of his long-time collaborator, Professor Luigi Salce. 4 surveys from top specialists stick with Professor Salce's article.They current fresh effects from lively examine parts: errors correcting codes as beliefs in crew earrings, duality in module different types, automorphism teams of abelian teams, and generalizations of isomorphism in torsion-free abelian teams. as well as those surveys, the quantity includes 22 learn articles in various parts attached with the subjects of the convention. The components mentioned contain abelian teams and their endomorphism earrings, modules over numerous earrings, commutative and non-commutative ring idea, sorts of teams, and topological points of algebra. The e-book bargains a entire resource for contemporary examine during this lively zone of analysis

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**Extra resources for Abelian Groups, Rings and Modules: Agram 2000 Conference July 9-15, 2000, Perth, Western Australia**

**Example text**

Zn is regular in R = R/(z1 , . . , z −1 ). But z +2 , . . , zn is regular in R = R/(z1 , z2 , . . , z , z +1 ), so the result follows from the previous paragraph. 8 Cohen-Macaulay Rings The depth(R) of a ring local ring (R, m) is deﬁned as the maximal length of a regular sequence in m. It can be shown that in a graded Noetherian ring R any two homogeneous maximal regular sequences have the same length. If R is a graded Noetherian ring we put depth(R) = depth(R+ ). The depth of a ring is always less than or equal to its Krull dimension.

F¯r } form a vector space basis of Q(R+ ). Then f1 , f2 , . . , fr minimally generate R as a K algebra. To see r r this consider that R+ = i=1 Rfi . Thus if f ∈ Rd then f = i=1 gi fi for some homogeneous elements gi ∈ R+ . By induction on degree, we see that gi ∈ K[f1 , f2 , . . , fr ] and thus f ∈ K[f1 , f2 , . . , fr ]. Conversely, it is clear that 2 an element of R not lying in R+ cannot be written as a polynomial in lower degree elements of R and thus any homogenous algebra generating set must surject onto to a spanning set for Q(R+ ).

Z , z +1 ), so the result follows from the previous paragraph. 8 Cohen-Macaulay Rings The depth(R) of a ring local ring (R, m) is deﬁned as the maximal length of a regular sequence in m. It can be shown that in a graded Noetherian ring R any two homogeneous maximal regular sequences have the same length. If R is a graded Noetherian ring we put depth(R) = depth(R+ ). The depth of a ring is always less than or equal to its Krull dimension. A local Noetherian ring (R, m) is said to be Cohen-Macaulay if depth(m) = height(m) and a general Noetherian ring R is said to be Cohen-Macaulay if the localization of R at each of its prime ideals is Cohen-Macaulay.

### Abelian Groups, Rings and Modules: Agram 2000 Conference July 9-15, 2000, Perth, Western Australia by Andrei V. Kelarev, R. Gobel, K. M. Rangaswamy, P. Schultz, C. Vinsonhaler

by Mark

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