Author Archives: limsup

Commutative Algebra 14

Basic Open Sets For , let , an open subset of Spec A. Note that . Proposition. The collection of over all forms a basis for the topology of . Proof Let be an open subset of Spec A. Suppose so … Continue reading

Posted in Advanced Algebra | Tagged , , , , , , , , | Leave a comment

Commutative Algebra 13

Zariski Topology for Rings In this article, we generalize earlier results in algebraic geometry to apply to general rings. Recall that points on an affine variety V correspond to maximal ideals . For general rings, we have to switch to … Continue reading

Posted in Advanced Algebra | Tagged , , , , , , | Leave a comment

Commutative Algebra 12

Some Results on Posets In this article we have two goals in mind: to introduce the idea of noetherian posets, and to state Zorn’s lemma and give some examples. The latter is of utmost importance in diverse areas of mathematics. … Continue reading

Posted in Advanced Algebra | Tagged , , , , | Leave a comment

Commutative Algebra 11

Coordinate Rings as k-algebras Let k be an algebraically closed field. Recall that a closed subset is identified by its coordinate ring k[V], which is a finitely generated k-algebra since Definition. An affine k-variety is a finitely generated k-algebra A which is … Continue reading

Posted in Advanced Algebra | Tagged , , , , , | Leave a comment

Commutative Algebra 10

Algebras Over a Ring Let A be any ring; we would like to look at A-modules with a compatible ring structure. Definition. An –algebra is an -module , together with a multiplication operator such that becomes a commutative ring (with 1); multiplication … Continue reading

Posted in Advanced Algebra | Tagged , , , , , , | Leave a comment

Commutative Algebra 9

Direct Sums and Direct Products Recall that for a ring A, a sequence of A-modules gives the A-module where the operations are defined component-wise. In this article, we will generalize the construction to an infinite collection of modules. Throughout this article, let denote … Continue reading

Posted in Advanced Algebra | Tagged , , , , | 2 Comments

Commutative Algebra 8

Generated Submodule Since the intersection of an arbitrary family of submodules of M is a submodule, we have the concept of a submodule generated by a subset. Definition. Given any subset , let denote the set of all submodules of M containing … Continue reading

Posted in Advanced Algebra | Tagged , , , , , , | Leave a comment