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Concept guide



## [Finite two-sided ideal: checking left and right absorption separately in an upper-triangular matrix ring](/en/derivations/derivation-mko-finite-two-sided-ideal-01/?from=%2Fen%2Fderivations%2F%3Fq%3Dmko-finite-two-sided-ideal)

The parent ring U is the ring of 2-by-2 upper-triangular matrices over 𝔽₂ (elements written (a,b,c) for the matrix with first row (a,b) and second row (0,c)), already qualified as a finite unital ring. (1) Prove that I={(0,0,0),(0,1,0)} is a two-sided ideal of U. (2) Prove that L={(0,0,0),(1,0,0)} satisfies left absorption but is not a two-sided ideal. (3) Prove that the diagonal subring D={(a,0,c)} contains 1 but is not an ideal. (4) Show that {0} and U itself are two-sided ideals.

Finite two-sided ideal

derivation-mko-finite-two-sided-ideal-01





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