Displaying similar documents to “When is a Ring of Torus Invariants a Polynomial Ring?”

σ-ring and σ-algebra of Sets1

Noboru Endou, Kazuhisa Nakasho, Yasunari Shidama (2015)

Formalized Mathematics

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In this article, semiring and semialgebra of sets are formalized so as to construct a measure of a given set in the next step. Although a semiring of sets has already been formalized in [13], that is, strictly speaking, a definition of a quasi semiring of sets suggested in the last few decades [15]. We adopt a classical definition of a semiring of sets here to avoid such a confusion. Ring of sets and algebra of sets have been formalized as non empty preboolean set [23] and field of subsets...

Isomorphism Theorem on Vector Spaces over a Ring

Yuichi Futa, Yasunari Shidama (2017)

Formalized Mathematics

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In this article, we formalize in the Mizar system [1, 4] some properties of vector spaces over a ring. We formally prove the first isomorphism theorem of vector spaces over a ring. We also formalize the product space of vector spaces. ℤ-modules are useful for lattice problems such as LLL (Lenstra, Lenstra and Lovász) [5] base reduction algorithm and cryptographic systems [6, 2].