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A Singular Introduction to Commutative Algebra
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- A Singular Introduction to Commutative Algebra.
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Views Total views. Actions Shares. Embeds 0 No embeds. No notes for slide. Detailed explanations of the various SINGULAR commands do not usually appear within the text, but in an appendix , so the rationale for some constructions might not be immediately apparent. Books with a computational flavor mix constructive approaches, in the form of algorithms and computations, with theorems and proofs.
A SINGULAR introduction to commutative algebra pdf - Web Education
Constructive approaches often reveal information hidden by elegant but abstract proofs; the authors of this text cite primary decomposition as a case where the constructive approach reveals a great deal more. In most textbooks, the computational flavor manifests itself in the form of algorithms written in pseudocode and applications to areas such as robotics or financial mathematics.
The general nature of pseudocode means that the reader can program the algorithm on any computer algebra system, so long as she knows how to translate pseudocode into that system's language. The drawback to this is that it can steepen the learning curve for students unaccustomed to programming.
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The focus on SINGULAR allows the authors to state precisely how the user should interact with the program and to provide procedures — not only pseudocode! The reader encounters right away in Section 1.
Local orderings are necessary for standard bases of ideals of power series rings and local rings, useful for the analysis of singularities. After a lengthy introduction to the theory of standard bases for rings and modules, the textbook treats a number of important topics in commutative algebra: Noether normalization, primary decomposition, the Hilbert function, and homological algebra.
The three substantial appendices account for one-third of the book's nearly pages, and include a foray into ideas of algebraic geometry related to topics in the text, a manual on SINGULAR, and a chapter on polynomial factorization.
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The writing is clear and precise. Most of the examples are excellent and insightful, although a few are a little trivial. John Perry is an assistant professor of mathematics at the University of Southern Mississippi.
His mathematical interests lie primarily in computational algebra. He once had a number interests outside of mathematics, but after three children and several home renovations he has forgotten what it means to have free time. Noether Normalization and Applications. Primary Decomposition and Related Topics. Hilbert Function and Dimension. Complete Local Rings.