Abstract Algebra Answers

(i) For non-trivial subspaces U and W of a (finite-dimensional) vector space V, define
U +W := {u + w | u element of U and w element of W}.
Prove that U +W is a subspace of V .
(ii) Show that
dim(U +W) = dim(U) + dim(W) − dim(U intersect W)
by considering a basis for U intersect W, extending it to bases for U and W, and then
identifying, with justification, a basis for U +W in terms of these elements.

Give an example of a polynomial of degree 100 over the rationals that is irreducible over the rationals

Let R be the commutative Q-algebra generated by x1, x2, . . . with the relations (x_n)^n= 0 for all n. Show that R does not have a largest nilpotent ideal.

Give an example to show that
rad(R) ⊆ {r ∈ R : r +U(R) ⊆ U(R)}.

need not be an equality.

It is well known that, for any commutative noetherian R, (intersection) (rad R)^n = 0. Show that this need not be true for noncommutative right noetherian rings.

Using the definition of rad R as the intersection of the maximal left ideals, show directly that rad R is an ideal.

Show that for any ideal I ⊆ rad R, the natural map GLn(R) → GLn(R/I) is surjective.

Construct an artinian ring R in which soc(R_R) is not equal soc(RR).

For any left artinian ring R with Jacobson radical J, show that
soc(R_R) = {r ∈ R : Jr = 0} and soc(RR) = {r ∈ R : rJ = 0}.

Give proof for the fact that if R is a simple ring which has a minimal left ideal, then R is a semisimple ring.