Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Eisenstein's criterion wikipedia , lookup
Fundamental theorem of algebra wikipedia , lookup
Dedekind domain wikipedia , lookup
Ring (mathematics) wikipedia , lookup
Gröbner basis wikipedia , lookup
Cayley–Hamilton theorem wikipedia , lookup
Polynomial ring wikipedia , lookup
Math 3101 Spring 2017 Homework 2 1. Let R be a unital ring and let I be a left ideal of R. Show that I = R if and only if I contains 1R . (Note that any ring is in fact a left, right, and two-sided ideal of itself; you do not have to prove this fact. Also note that the statement in the problem remains true if I is instead assumed to be a right ideal or a two-sided ideal.) 2. Let R be a unital ring, let n be a positive integer, and let Mn (R) denote the unital ring of n×n matrices with entries in R. (You do not have to prove that Mn (R) is a unital ring.) (a) Let I be an ideal of R. Prove that a b Mn (I) = a, b, c, d ∈ I c d is an ideal of Mn (R). (b) Let J be an ideal of Mn (R), and let E(J) denote the subset of R comprised of entries of matrices in J. Prove that E(J) is an ideal of R, and that J = Mn (E(J)). (Hint: Consider the elementary matrices eij for 1 ≤ i, j ≤ n.) Conclude that every ideal of Mn (R) has the form Mn (I) for some choice of ideal I of R. (c) Let K be a field. Conclude from the above that the only ideals of Mn (K) are the zero ideal and Mn (K) itself. 3. Let K be a field, and let R be the ring of 2×2 matrices with entries in K. Find a left ideal of R that is not a right ideal and a right ideal of R that is not a left ideal. Justify your answers. 4. Let K be a field, and let R be the set of upper triangular 2×2 matrices with entries in K. That is, R is comprised of all matrices of the form a b 0 c for a, b, c ∈ K. (a) Prove that R is a unital subring of M2 (K). (b) Find five distinct two-sided ideals of R. Justify your answer. (Can you prove that this list is complete?) 5. Let X be a nonempty set and let R be a ring. Recall the set F (X, R) of functions from X to R, and recall that F (X, R) is a ring under pointwise addition and multiplication. (You do not have to prove that F (X, R) is a ring.) Note, for every r ∈ R, that we have the constant function fr with fr (x) = r for all x ∈ X. (a) Prove that the set of constant functions in F (X, R) forms a subring of F (X, R). (b) Prove that the subring of constant functions of F (X, R) is isomorphic to R itself.