I'm not sure what you mean by the phrase after the comma. Clearly finite-dimensional algebras are easier to study than more general rings. They are a smaller class of objects and we have the techniques of linear algebra available as tools. But one of the ways in which mathematics progresses is through generalisation, and results about algebras are often generalised to wider classes of rings.
Thanks, that's true. I was thinking more along the lines of a notion like Morita equivalence (or a weakened notion) being restricted from rings to finite-dimensional algebras, not to mention enhanced to operator algebras. It retained its name and citation, even though the Americans and Europeans might have ignored it for very strict mathematical reasons or for political reasons such as Japan was a defeated nation after WWII.
The specialization of ring notions into algebras has been already quite explored. Since Ring Theory is quite a wild world, the specialization to a more restricted context becomes reacher and has its own developments. I am familiar with both, the setting of operator algebras and the one of finite dimensional algebras, and certainly ring theory is used.
As suggested by David Towers, the more restricted context of algebras also produces new ideas that had been exported to the general setting of Ring Theory. It is particularly interesting the case of Morita equivalence and weakened notions, in particular, the case o of Tilting equivalence.
A Tilting equivalences is a weak version of Morita equivalence, born in the setting of classification theory of finite dimensional algebras as an abstract version of Berenstein-Gelfand-Ponomarev Coxeter functors. Tilting equivalences and tilting modules have grown to applications in general ring and module theory, homological algebra and algebraic geometry.
Yes I agree with professor @D.Herbera and as he said it is a wild word and its very hard for me to choosing for search in a special place in this word. All part of this word was interested
While seconding the comments by David A. Towers , and Dolors Herbera , I like to add a comment too. If we restrict to the ring theory by just the abstract definition of rings, we are are not able to obtain any nontrivial result whatsoever. Even the fact that every ideal of a ring with unity is contained in a maximal ideal is not a ring theoretical result, it is indeed a fact in set theory (it's equivalent to the Axiom of Choice). When we present a definition in mathematics we should first of all show that there are some objects satisfying our definition. Then point out to some of these objects as important prototypes of these objects, which already exist and been studied for years in mathematics. Finally, we should try to prove or extend some of the properties of these important prototypes to our general object that we have already defined.
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