Commutative rings I

By Paul B. Garrett. 2007

704 citations recorded by the source corpus.

View Paper: https://www.semanticscholar.org/paper/b8ccc99e527d93135d844e4ea43017ee9d98905a

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Citations in the corpus

Zero-Divisor Graphs of Matrices Over Commutative Rings (2009, background): “We have by Theorem 82 in Kaplansky (1974) that Z R is annihilated by a single element, say a.”
Direct sums of Rickart modules (2012, unknown): “Hence by Theorem 63 in [16], the localization RP is a valuation domain for each prime ideal P of R.”
Properties of chains of prime ideals in an amalgamated algebra along an ideal (2010, background): “Anderson in [1] and motivated by a classical construction due to Dorroh [12], concerning the embedding of a ring without identity in a ring with identity.”
The total graph of a commutative ring (2008, background): “General references for ring theory are [10] and [11].”
The image of Colmez’s Montreal functor (2010, background): “Since R[1/p] is a field and R is a noetherian integral domain Theorem 146 in [37] implies that R/pR is artinian.”
Unit Graphs Associated with Rings (2010, methodology): “We use the notation of Kaplansky [9] and Chartrand and Oellermann [3].”
F-signature and Hilbert–Kunz multiplicity : a combined approach and comparison (2016, unknown): “Again, U 1 i J ( n ) is principal (as U 1 i R is a semilocal domain, every locally principal ideal is principal by [Kaplansky 1974, Theorem 60]), so we can choose x i ∈ U i and a i ∈ J ( n ) so that J ( n ) x i = a i R x i and x ni J ( n ) ⊆ a i R .”
Zero-Divisor Graph with Respect to an Ideal (2006, background): “2000 Mathematics Subject Classification: 05C75; 13A15.”
ON THE DIAMETER AND GIRTH OF A ZERO-DIVISOR GRAPH (2007, background): “For any undefined ring-theoretic terminology, see [9] or [10].”
Zero-divisor graphs of amalgamated duplication of a ring along an ideal (2007, background): “The concept of thegraph of the zero divisors ofR was first introduced by Beck [4], where he was mainly interested in colorings.”
THE GENERALIZED TOTAL GRAPH OF A COMMUTATIVE RING (2013, unknown): “General references for ring theory are [18, 19].”
On the Zero-Divisor Graph of a Ring (2008, background): “For any undefined ringtheoretic concepts or terminology, see Huckaba (1988) or Kaplansky (1974).”
Idempotent and Nilpotent Submodules of Multiplication Modules (2008, background): “Address correspondence to Majid M. Ali, Department of Mathematics, College of Science, Sultan Qaboos University, P.O. Box 36, 123 Al Khodh, Muscat, Oman; Fax: +968-2414145; E-mail: mali@squ.edu.om 4620 R or zero (Kaplansky, 1974).”
Rings with Property (A) and their extensions (2007, methodology): “One of important properties of commutative Noetherian rings is that the annihilator of an ideal I consisting entirely of zero-divisors is nonzero [12, p. 56].”