Algebra I. Homework 4. Due on October 8, 2020. 1. Prove or disprove: a subring of a Noetherian ring is Noetherian. 2. Let p be a prime number. Show that the polynomial 1 + x + . . . + x

In particular, every associated prime ideal of M is maximal if and only if M is locally of finite length, in the sense that every finitely generated submodule of M has finite

Suppose that G is a finitely generated semi- group such that every action by Lipschitz transformations (resp.. First it must be checked that the action is well-defined. The first one

Denote by H 0 (X, L) the vector space of global sections of L, and suppose that this space has

Construct a second family of lines contained in

Let us denote by RG perm p the category of all p {permutation RG {modules. The following proposition generalizes to p {permutation modules a result which is well known for

If a real number x satisfies Conjecture 1.1 for all g and a, then it follows that for any g, each given sequence of digits occurs infinitely often in the g-ary expansion of x.. This

In the mid-1980s the second author observed that for a connected reductive complex algebraic group G the singular support of any character sheaf on G is contained in a fixed

(15) (*) A topological space is quasi-compact if one can extract from any open covering a finite open covering.. Show that a Noetherian topological space