Let R be a commutative von Neumann regular ring. We show that every finitely
generated ideal I in the ring of polynomials R[X] has a strong Gr?bner
basis. We prove this result using only the defining property of a von
Neumann regular ring.
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The conditions that allow an element of an associative, unital, not necessarily commutative ring R, to be represented as a sum of (commuting) idempotents and one nilpotent element are analyzed. Some applications to group rings are also presented.
Let [Formula: see text] be an associative unital ring and not necessarily commutative. We analyze conditions under which every [Formula: see text] matrix [Formula: see text] over [Formula: see text] is expressible as a sum [Formula: see text] of (commuting) idempotent matrices [Formula: see text] and a nilpotent matrix [Formula: see text].
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