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The main purpose of the present paper is to estimate id-A a, the injective dimension of A a, in terms of those of RR, MR, and Ss. In fact,if we assume that fd-sM, the flatdimension of SM, is finite,then there hold the inequalities max(id-RR, id-MR, id-Ss-fd-sM)^id-AA^ max (max (id-RR, id-MR)+fd-sM, id-S5-l)+l. In this connection, we investigate the case when the left-hand or the righthand side equality holds under the condition that SM is flat. In [7], Zaks shows that the injective dimension of an nXn lower triangular matrix ring over a semiprimary ring R is just equal to id-RR-}-l. An example is constructed to show that the condition on R benig semiprimary is redundant in his theorem. The author wishes to express his hearty thanks to Professors H. Tachikawa and T. Kato for their useful suggestions and remarks. Let ,=[J

本研究では完全代数的独立性および微分完全代数的独立性という著しい性質を有する関数を、代表的な非アルキメデス付値体の上で構成した。初年度には、実数であると同時に有限個の素数pに対するp進数でもあり、有理数体上で代数的独立となる無限集合の実例を得た。その後、完全代数的独立性の拡張概念を得てp進数体において完全代数的独立性を有する関数を構成する基盤を築いた。また、正標数の関数体上で微分完全代数的独立性を有する関数を、多変数Mahler関数を用いて構成した。最終年度には、微分完全代数的独立性を有する関数を応用し超越性・代数的独立性を行列環上に拡張する新たな概念を得た。 In this project the research representative constructed certain functions having ultimate algebraic independence properties especially over typical non-Archimedean valuation fields. First, he obtained the infinite algebraically independent sets consisting of numbers which can be regarded both as real and as p-adic for a finite number of primes p. Then he established the base for constructing functions having perfect algebraic independence properties over p-adic number fields. Over function fields of positive characteristic, the research representative constructed, using Mahler functions of several variables, the functions having differential perfect algebraic independence properties. Finally, he extended the concept of transcendence and algebraic independence to matrix rings as applications of the functions having differential perfect algebraic independence properties., 研究種目 : 基盤研究 (C) (一般) 研究期間 : 2015～2019 課題番号 : 15K04792 研究分野 : 数物系科学