Why do we usually quotient a polynomial ring by a monic irreducible polynomial? I'm studying some ring theory, and I'm wondering for a polynomial ring $F[x]$ over a field $F$, if we quotient by some element $f(x) \in F[x]$ to get $F[x]/(f(x))$, why do we usually want $f(x)$ to be monic irreducible? How about arbitrary polynomials $g(x)$?
Quotient by irreducible to get an integral domain because if it is not irreducible say $F[x]/g(x)f(x)$ then $f(x),g(x)$ are non zeros element in the quotient ring such that $f(x)g(x)=0$
Quotient by irreducible to get an integral domain because if it is not irreducible say $F[x]/g(x)f(x)$ then $f(x),g(x)$ are non zeros element in the quotient ring such that $f(x)g(x)=0$