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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?

https://brainmass.com/math/algebra/submodules-polynomial-ring-kx-564383

#### Solution Preview

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

#### 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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