Answer to Question #47347 in Abstract Algebra for ayush kumar

Question

(a) If H and K are subgroups of a group G, and 4 if only H is a normal subgroup of G, then prove or disprove that HK is a subgroup of G. Give an example to show that HK need not be a subgroup if neither H nor K is a normal subgroup of G.

(b) Show that (I) : R[x]-->R, defined by ((ao + + + a nxn) ao + + + an, is a ring homomorphism. Check whether ker (I) is a principal ideal or not. Is it a maximal ideal ? Why, or why not ?