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

Answer 1

Avery Alexander

A ring with zero divisors is necessarily non-commutative.

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Answer 2

Genesis Allen

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

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.

As an illustration:

The number((1,0),(0,0))((0,0),(0,1)) equals ((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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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.