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Zariski topology. (English) Zbl 1422.13005

Summary: We formalize in the Mizar system basic definitions of commutative ring theory such as prime spectrum, nilradical, Jacobson radical, local ring, and semi-local ring, then formalize proofs of some related theorems along with the first chapter of [M. F. Atiyah and I. G. Macdonald, Introduction to commutative algebra. Reading, MA etc.: Addison-Wesley Publishing Company (1969; Zbl 0175.03601)].
The article introduces the so-called Zariski topology. The set of all prime ideals of a commutative ring \(A\) is called the prime spectrum of \(A\) denoted by Spectrum \(A\). A new functor Spec generates Zariski topology to make Spectrum \(A\) a topological space. A different role is given to Spec as a map from a ring morphism of commutative rings to that of topological spaces by the following manner: for a ring homomorphism \(h:A\rightarrow B\), we defined \((\operatorname{Spec}h):\operatorname{Spec}B\rightarrow\operatorname{Spec}A\) by \((\operatorname{Spec}h)(\mathfrak{p})=h^{-1}(\mathfrak{p})\) where \(\mathfrak{p}\in \operatorname{Spec}B\).

MSC:

13A15 Ideals and multiplicative ideal theory in commutative rings
03B35 Mechanization of proofs and logical operations

Citations:

Zbl 0175.03601

Software:

Mizar
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Full Text: DOI

References:

[1] Michael Francis Atiyah and Ian Grant Macdonald. Introduction to Commutative Algebra, volume 2. Addison-Wesley Reading, 1969.; · Zbl 0175.03601
[2] Jonathan Backer, Piotr Rudnicki, and Christoph Schwarzweller. Ring ideals. Formalized Mathematics, 9(3):565-582, 2001.;
[3] Grzegorz Bancerek, Czesław Byliński, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, Karol Pąk, and Josef Urban. Mizar: State-of-the-art and beyond. In Manfred Kerber, Jacques Carette, Cezary Kaliszyk, Florian Rabe, and Volker Sorge, editors, Intelligent Computer Mathematics, volume 9150 of Lecture Notes in Computer Science, pages 261-279. Springer International Publishing, 2015. ISBN 978-3-319-20614-1. doi:10.1007/978-3-319-20615-8_17.; · Zbl 1417.68201
[4] Grzegorz Bancerek, Czesław Byliński, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, and Karol Pąk. The role of the Mizar Mathematical Library for interactive proof development in Mizar. Journal of Automated Reasoning, 61(1):9-32, 2018. doi:10.1007/s10817-017-9440-6.; · Zbl 1433.68530
[5] Shigeru Iitaka. Algebraic Geometry: An Introduction to Birational Geometry of Algebraic Varieties. Springer-Verlag New York, Inc., 1982.; · Zbl 0491.14006
[6] Shigeru Iitaka. Ring Theory (in Japanese). Kyoritsu Shuppan Co., Ltd., 2013.; · Zbl 0656.14019
[7] Christoph Schwarzweller. The binomial theorem for algebraic structures. Formalized Mathematics, 9(3):559-564, 2001.;
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