Wikipedia on local rings calls a local ring any ring with a unique maximal left or right ideal.
Up until now, I used to privately call these rings left local and right local respectively (realizing these notions are equivalent) and assumed that local rings were rings with a unique maximal two-sided ideal. This would include simple rings that aren’t skew fields as local rings – so for example matrix rings over skew fields.
- Why does the correct generalization use one-sided ideals?
- Wouldn’t it make sense to just speak of local modules in the general case?
It is not the case that the Jacobson radical is the intersection of all maximal two sided ideals. See for example
https://mathoverflow.net/questions/50360/jacobson-radical-intersection-of-all-maximal-two-sided-ideals
https://mathoverflow.net/questions/775/what-is-an-example-of-a-ring-in-which-the-intersection-of-all-maximal-two-sided-i
So the condition that there is a unique maximal two sided ideal is not really generalizing the property that we want: that there is a unique simple module (up to isomorphism).
One could of course ask that the ring modulo the Jacobson radical is simple artinian, so a matrix ring over a skew field, but then one runs in to the problem that one cannot in general lift idempotents modulo the Jacobson radical (this we can do in semiperfect rings), so our ring will not necessarily be Morita equivalent to a local ring.