# Concept derivation guides

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

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Concept guide



## [Finite commutative ring: checking commutativity of ℤ₆ over all element pairs, with an upper-triangular matrix ring as a non-commutative counterexample](/en/derivations/derivation-mko-finite-commutative-ring-01/?from=%2Fen%2Fderivations%2F%3Fq%3Dmko-finite-commutative-ring)

(1) On ℤ₆={0,1,2,3,4,5} (a⊕b=(a+b) mod 6, a⊗b=(ab) mod 6), already qualified as a finite unital ring, check a⊗b=b⊗a for all 36 ordered element pairs and decide whether ℤ₆ is a finite commutative ring. (2) Show that ℤ₆ has nonzero zero divisors and that 2 is not invertible, so it is neither an integral domain nor a field. (3) In the qualified ring U of 2-by-2 upper-triangular matrices over 𝔽₂, show that several products commute and yet U is not commutative.

Finite commutative ring

derivation-mko-finite-commutative-ring-01





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