I What's the Difference Between Zero Divisors and Torsion Elements?

[CSteiner]

So I've been studying advanced linear algebra and have started learning about modules. However, I am having a hard time understanding the difference between a zero divisor and a torsion elements. The definitions seem extremely similar. Can someone offer a good definition of each and an explanation of the difference?

 

[mathwonk]

i looked in atiyah macdonald and found this.

zero divisor is a concept applied to a ring, while torsion element is the analogous concept applied to a module. i.e. in a ring A, x is a zero divisor if xy=0 for some y≠0 in A.

If A is a ring and M is an A module, an element z of m is a torison element if xz=0 for some non zero divisor x in A.e.g. in an abelian group M, hence a module over the integers, an element z of M is torsion if and only if it generates a subgroup of finite order, i.e. if and only if nz = 0 for some n ≠0. in the integers there are of course no zero divisors (except 0).

 

[CSteiner]

Okay I think I understand now. However, I am noticing that the first definition invokes right multiplication whereas the second definition invokes left multiplication. In a module over a commutative ring this obviously won't make a difference, but is there some sort of subtle significance in the non commutative case?

 

[mathwonk]

i was thinking of the commutative case as i always do. (i read it in atiyah macdonald's book "commutative algebra".) in the non commutative case i suppose one has more notions, left zero divisor, right zero divisor, and a torsion element should be annihilated by a ring element which is neither i suppose, i.e. a "regular" element, but i am not an expert.

 

[CSteiner]

Okay, that makes sense, thanks!

 

[mathwonk]

there seems however to be some variation among different authors as Dummit and Foote e.g. define torsion elements more broadly, as any z such that xz=0 for a non zero element x of the ring, without requiring that x is not zero divisior. the more restriced definition came from a wiki article i found.