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Tensor products Let R be a commutative ring. Given R-modules M1 , M2 and N we say that a map b: M1 × M2 → N is R-bilinear if for all r, r0 ∈ R and module elements mi , m0i ∈ Mi we have b(r · m1 + r0 · m01 , m2

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Pye Phyo Aung* (). Gorenstein Dimensions, Amalgamated Duplication of a Ring, and Pseudocanonical Covers. When E is a semidualizing module over a commutative ring, Holm and Jørgensen studie

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EXTERIOR POWERS KEITH CONRAD 1. Introduction Let R be a commutative ring. Unless indicated otherwise, all modules are R-modules and all tensor products are taken over R, so we abbreviate ⊗R to ⊗. A bilinear function

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Finitely generated modules over a PID Lemma. Let R be a commutative ring with 1 6= 0. If Rm ∼ = Rn as R-modules then m = n. Proof. Let m ⊂ R be a maximal ideal, and let k = R/m. Then Rm ∼ = Rn implies that

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Young Women in Topology Bonn: June 25 – 27, 2010 Universal Toda brackets of Commutative Ring Spectra Boryana Dimitrova

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Natural Algorithms ∗ Bernard Chazelle Abstract We provide further evidence that the study of complex self-organizing systems can benefit from an algorithmic perspective. The subject has been traditionally viewed

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• Algebra
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• Nuclear magnetic resonance
• Relaxation
• Spectrum of a ring

TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 364, Number 8, August 2012, Pages 3967–3992 SArticle electronically published on March 22, 2012

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• Algebra
• Abstract algebra
• Mathematics
• Semigroup theory
• Algebraic structures
• Ring theory
• Commutative algebra
• Monoid
• Semigroup
• Cancellation property
• Inverse element
• Integral domain

MATHEMATICS OF COMPUTATION SArticle electronically published on February 15, 2002 CLASS NUMBERS OF REAL CYCLOTOMIC FIELDS OF PRIME CONDUCTOR

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• Algebra
• Abstract algebra
• Mathematics
• Commutative algebra
• Algebraic structures
• Ring theory
• Homological algebra
• Algebraic number theory
• Algebraic number field
• Divisor
• Gorenstein ring
• Ring