If a ring has zero divisors, is it necessarily commutative or non-commutative?

1 Answer
George C.
Aug 27, 2016

A ring can have zero divisors whether or not it is commutative.

Explanation:

Consider arithmetic modulo #4#, i.e. #ZZ_4# plus multiplication modulo #4#.

This is a commutative ring with #2# being a zero divisor.

#color(white)()#
The ring of #2xx2# matrices over any ring is a non-commutative ring with zero divisors.

For example:

#((1,0),(0,0))((0,0),(0,1)) = ((0,0),(0,0))#

#((1,1),(0,0))((1,0),(0,0)) = ((1,0),(0,0)) != ((1,1),(0,0)) = ((1,0),(0,0))((1,1),(0,0))#

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