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Quotient ring is also know as factor ring but what has it got to do with 'division' in any remote sense whatsoever? I know it is not meant to be division per se but why give the name of this ring the quotient ring or factor ring? What is the motivation behind it?

R/I={r in R| r+I}

Normally when we divide by something or obtain a quotient of something in ordinary arithematic, the quotient is simpler than the numerator, the thing we are dividing by. But in this case the quotient ring is more complicated than the original ring R in that the quotient ring is a set of sets where as R was a set of elements.

The same issue goes for quotient groups. It would make more sense if they were called addition rings instead of quotient rings and multiplication rings instead quotient groups since we are really adding and multiplying (although multiplication can be addition in groups) respectively. However that may create some confuction because the words addition and multiplication are too common already.

R/I={r in R| r+I}

Normally when we divide by something or obtain a quotient of something in ordinary arithematic, the quotient is simpler than the numerator, the thing we are dividing by. But in this case the quotient ring is more complicated than the original ring R in that the quotient ring is a set of sets where as R was a set of elements.

The same issue goes for quotient groups. It would make more sense if they were called addition rings instead of quotient rings and multiplication rings instead quotient groups since we are really adding and multiplying (although multiplication can be addition in groups) respectively. However that may create some confuction because the words addition and multiplication are too common already.

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