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Submodules of polynomial ring K[x]

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Which of the following sets of polynomials n K[x], K commutative are K-modules:(a) all polynomials of degree exactly ; (b) all polynomials of degree at most 4; (c) all monic polynomials; (d) all polynomials of even degree?

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Solution Summary

This question is on identifying whether a given subset of polynomial ring K[x] is also a K-module.

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Recall (M, +) is called an R-module if there is an operation • : R × M → M
such that
(1) r • (x + y) = r • x + r • y (2) (r + s) • x = r • x + s • x (3) (rs) • x = r • (s • x)
(4) 1R • x = x
For all r, s ∈ R, and x, y ∈ M .
Now let the sets in part (a) to (d) be A, B, C, D respectively.
(a) A is not K -module: because 0K • x ...

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