Thesis (Ph. D.)--University of Chicago, Dept. of Mathematics, June 1997. / Includes bibliographical references. Also available on the Internet.

Thesis (Ph. D.)--University of Chicago, Dept. of Mathematics, June 2000. / Includes bibliographical references. Also available on the Internet.

by Lok Tsan-ming. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1994. / Includes bibliographical references (leaves 74-75). / Introduction --- p.2 / Chapter 1 --- "Rings, Modules and Categories" --- p.4 / Chapter 1.1 --- "Rings, Subrings and Ideals" --- p.5 / Chapter 1.2 --- Modules and Categories --- p.8 / Chapter 1.3 --- Module Theory --- p.13 / Chapter 2 --- Isomorphisms between Endomorphism rings of Quasiprogener- ators --- p.24 / Chapter 2.1 --- Preliminaries --- p.24 / Chapter 2.2 --- The Fundamental Theorem --- p.31 / Chapter 2.3 --- Isomorphisms Induced by Semilinear Maps --- p.41 / Chapter 2.4 --- Isomorphisms of General linear groups --- p.46 / Chapter 3 --- Endomorphism ring of projective module --- p.54 / Chapter 3.1 --- Preliminaries --- p.54 / Chapter 3.2 --- Main Theorem --- p.60 / Bibliography --- p.74

Let <i>R</i> be a commutative ring, <i>I</i> be an ideal in <i>R</i> and let <i>M</i> be a <i>R/ I</i> -module. In this thesis we construct a <i>R/ I</i> -projective resolution of <i>M</i> using given <i>R</i>-projective resolutions of <i>M</i> and <i>I</i>. As immediate consequences of our construction we give descriptions of the canonical maps Ext_R/I<i>(M,N)</i> -> Ext_R<i>(M,N)</i> and Tor^R_N<i>(M, N)</i> -> Tor^R/I_n<i>(M, N)</i> for a <i>R/I</i> module <i>N</i> and we give a new proof of a theorem of Gulliksen [6] which states that if <i>I</i> is generated by a regular sequence of length r then ∐∞_n=o Tor^R/I_n <i>(M, N)</i> is a graded module over the polynomial ring </i>R/ I</i> [X₁. .. X_r] with deg X_i = -2, 1 ≤ i ≤ r. If <i>I</i> is generated by a regular element and if the <i>R</i>-projective dimension of <i>M</i> is finite, we show that <i>M</i> has a <i>R/ I</i>-projective resolution which is eventually periodic of period two. This generalizes a result of Eisenbud [3]. In the case when <i>R</i> = (<i>R</i>, m) is a Noetherian local ring and <i>M</i> is a finitely generated <i>R/ I</i> -module, we discuss the minimality of the constructed resolution. If it is minimal we call (<i>M, I</i>) a Golod pair over <i>R</i>. We give a direct proof of a theorem of Levin [10] which states thdt if (<i>M,I</i>) is a Golod pair over <i>R</i> then (Ωⁿ_R/IR/I(M),I) is a Golod pair over <i>R</i> where Ωⁿ_R/IR/I(M) is the nth syzygy of the constructed <i>R/ I</i> -projective resolution of <i>M</i>. We show that the converse of the last theorem is not true and if (Ω¹_R/IR/I(M),I) is a Golod pair over <i>R</i> then we give a necessary and sufficient condition for (<i>M, I</i>) to be a Golod pair over <i>R</i>. Finally we prove that if (<i>M, I</i>) is a Golod pair over <i>R</i> and if a ∈ <i>I</i> - m<i>I</i> is a regular element in </i>R</i> then (<i>M</i>, (a)) and (1/(a), (a)) are Golod pairs over <i>R</i> and (<i>M,I</i>/(a)) is a Golod pair over <i>R</i>/(a). As a corrolary of this result we show that if the natural map π : <i>R</i> → <i>R/1</i> is a Golod homomorphism ( this means (<i>R</i>/m, <i>I</i>) is a Golod pair over <i>R</i> ,Levin [8]), then the natural maps π₁ : <i>R</i> → <i>R</i>/(a) and π₂ : <i>R</i>/(a) → <i>R/1</i> are Golod homomorphisms. / Ph. D.