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Associates (Ring Theory)

I've encountered two definitions of what it means for two elements r, sr,\, s of a ring RR to be associates:

1.

∃u\exists u (a unit) st r=su r=su

2.

∃v\exists v (a unit) st r=vs r=vs

Several proofs in my current course in representation theory have assumed these to be equivalent, but I can't see why this is. (For example, using Defn. 1, it is claimed that if r and s are associates, then Rr=RsRr=Rs ; RR is not assumed to be commutative.)

Can anyone shed some light?
After a couple of hopeless efforts to prove this, I came up with the following.

(0010)=(0001)(0110)\begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix} = \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}

and yet

(0010)=(abcd)(0001)\begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix} = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}

has no solutions, let alone units.
Reply 2
Avatar for meatball893
meatball893
OP
11
Aaaaahhh, I'm a fool! This was being done in the context of Euclidean Domains, which are (apparently) commutative! :frown: Sorry for wasting your time, but thanks for your reply.

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