I'm suggesting to make noncommutative-rings a synonym to noncommutative-algebra, in the same way as commutative-rings has been made to ac.commutative-algebra.

Initially asked there, I'm asking it here to make it more visible. One moderator considered this request and replied that the tag descriptions are distinct and that it means that "somebody thinks the two are slightly different". For information, here are the two tag descriptions (2019/01/20):

- noncommutative-rings
*Questions about rings that are not necessarily commutative.* - noncommutative-algebra
*Non-commutative rings and algebras, non-associative algebras. Can be used in combination with*ra.rings-and-algebras.

My feeling is rather is that it was written by two different people and/or at two different times with no consideration to the other tag. Since it's hidden deeply in a long thread, it's not likely to be discussed by many people there. Whence my question:

is there any clear distinction between the two tags? If so, what would you suggest to clearly distinguish the tag descriptions, which would not be at odds with the current use of these tags? do you agree or disagree with this merging for any reason?

One thing is that "algebra" is more likely to be used when there's a ground field (even if rings in general are the same as $\mathbf{Z}$-algebras). In the commutative setting, it has been agreed that this doesn't make a sensible thematic distinction.

*Footnote 1.*Copy of my request (2018/02/18) as answer to the Help cleanup tags! thread:

The tags noncommutative-algebra and noncommutative-rings should be merged; the first being more used, and also in analogy with the tag ac.commutative-algebra, I'd suggest to embed noncommutative-rings into noncommutative-algebra.

Making a synonym noncommutative-rings $\to$ noncommutative-algebra would also have the advantage that the latter tag is suggested when people type "ring". (Note: commutative-rings has already been embedded in ac.commutative-algebra.)

*Footnote 2.*Related: On the top-level tag rings-and-algebra.