End n(k) is the set of all polynomial mappings: k^n->k^n. i have to prove that end n(k) is a monoid.(adsbygoogle = window.adsbygoogle || []).push({});

k is a field of q elements and n is the number of variables.

the composition of two mappings F G is: F o G = F o G(v) = [F1(G1(v),..Gn(v)), ... Fn(G1(v),...Gn(v))]

i must prove that the composition is associative and that there is a unit element.

id=X=(x1,...,xn) is obviously the unit.

but how do i prove associativity? i was thinking of using commutativity somehow:(x^n)^m = x^(n*m) = x^(m*n) = (x^m)^n

last, how do i show that End n(k) is not a ring? is it distributivity it fails? for all x,y,z E A ->

(x+y)z=xz + yz og z(x+y)=zx + zy

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# Prove that a set is a monoid, but not a ring.

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