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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 Preview

    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 ...

    Solution Summary

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