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Let $R$ be a unital ring (not necessarily commutative) and $x,y\in R$ be non-invertible elements. Then is $xy$ non-invertible? And how about the case $xy=yx$?

(postsclipt) Can the product of two non invertible elements in a ring be invertible? : This page gives a counterexample of the case $xy\neq yx$.

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marked as duplicate by Inactive - avoiding CoC, Parcly Taxel, Lord Shark the Unknown abstract-algebra Nov 1 '18 at 6:34

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  • $\begingroup$ It's exactly as you said. Then how is the latter case? $\endgroup$ – Ichiko Oct 31 '18 at 23:56

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