I Understanding Torsion Elements

1. Apr 22, 2016

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?

2. Apr 22, 2016

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).

Last edited: Apr 22, 2016
3. Apr 22, 2016

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?

4. Apr 22, 2016

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.

5. Apr 22, 2016

CSteiner

Okay, that makes sense, thanks!

6. Apr 22, 2016

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.