# What are the generalized statements of given statements?

Let $R$ be a ring and $M$ be an $R$-module.

Thm1. If $R$ is a division ring, then $M$ is free

Thm2. If $R$ is commutative, the rank of $M$ is unique.

First of all, are these statements true? (There's no reference, I just formulated by guessing)

Secondly, what is the generalized statement of the above theorems?

• Thm1 is true, see e.g. math.stackexchange.com/q/75866/15416, Thm2 is in general false, en.wikipedia.org/wiki/Invariant_basis_number. What do you mean by "generalized statement"? Do you mean a common generalisation of those two theorems? – Julian Kuelshammer Mar 6 '15 at 20:12
• @JulianKuelshammer I was asking generalization for each theorem and invariant basis number is what I was looking for. By the way, the link you gave says that a commutative ring is IBN, hence the rank is unique. Why Thm2 is false in general? – Rubertos Mar 6 '15 at 20:14
• Sorry, my mistake. – Julian Kuelshammer Mar 6 '15 at 20:18
• Artin-Wedderburn Theorem can be viewed as a generalisation of Thm1: en.wikipedia.org/wiki/Artin%E2%80%93Wedderburn_theorem – Julian Kuelshammer Mar 6 '15 at 20:19
• @JulianKuelshammer Thank you :) Would you please write that as an answer? – Rubertos Mar 6 '15 at 20:37