# Concept derivation guides

299 concept teaching guides, separate from the 36,000 individual question derivations.

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Concept guide



## [Polynomial ring over a finite commutative ring: addition, convolution and degree checks in (ℤ/4ℤ)[x]](/en/derivations/derivation-mko-polynomial-ring-over-finite-commutative-ring-01/?from=%2Fen%2Fderivations%2F%3Fq%3Dmko-polynomial-ring-over-finite-commutative-ring)

Over R=ℤ/4ℤ take f=1+2x and g=2x. Using the coefficient-sequence definition, compute and normalize f+g, f·g, f·f and g·g; prove the top-coefficient formula (fg)_{m+n}=f_m g_n and explain why deg(fg)=deg f+deg g fails here; finally prove that R[x] is not a finite ring.

Polynomial ring over a finite commutative ring

derivation-mko-polynomial-ring-over-finite-commutative-ring-01





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