Let R and S be rings, F a field, and f : R → S, g : F → R ring homomorphims. (a) Prove that, for any r ∈ R and n ∈ Z, f (nr) = nf (r). (b) Prove that for c ∈ F with c ̸= 0, if f (c) = 0R, then f is the zero map, i.e. for all x ∈ F , f (x) = 0R. (c) Prove that f if f is not the zero homomorphism, then f is injective.

Show that if [E:F]=2, then E is a splitting field over F.

(5) Recall that an element of a ring is idempotent if a² = a. (a) Show that any integral domain can have only 1 and 0 as idempotent elements. (b) Is the converse to the previous statement true? Hint: Consider the ring of continuous functions defined on the interval [0, 1], which we call C([0, 1]). What are the idempotent elements? Is this ring an integral domain?

Let E/F be an extension of fields. If [E: F] = 3 and 7 € E, 7 & F, show that the set V = {a+by|a,b € F} is not a subfield of E. What happens when Y EF?

Abstract algebra 2. Graduate level. Note: let R be any ring and M and N R − modules