If
I is a ring element of
R or
ZZ, or a list or sequence of such elements, then
I is understood to be the ideal generated by these elements. If
I is a module, then it must be a submodule of a free module of rank 1.
i1 : ZZ[x]/367236427846278621
ZZ[x]
o1 = 
367236427846278621
o1 : QuotientRing

i2 : A = QQ[u,v];

i3 : I = ideal random(A^1, A^{2,2,2})
9 2 1 9 2 1 2 3 2 3 2 3 7 2
o3 = ideal (u + u*v + v , u + u*v + v , u + u*v + v )
2 2 4 2 4 2 4 4
o3 : Ideal of A

i4 : B = A/I;

i5 : use A;

i6 : C = A/(u^2v^2,u*v);

i7 : D = GF(9,Variable=>a)[x,y]/(y^2  x*(x1)*(xa))
o7 = D
o7 : QuotientRing

i8 : ambient D
o8 = GF 9[x, y]
o8 : PolynomialRing

The names of the variables are assigned values in the new quotient ring (by automatically running
use R) when the new ring is assigned to a global variable.
Warning: quotient rings are bulky objects, because they contain a GrÃ¶bner basis for their ideals, so only quotients of ZZ are remembered forever. Typically the ring created by R/I will be a brand new ring, and its elements will be incompatible with the elements of previously created quotient rings for the same ideal.
i9 : ZZ/2 === ZZ/(4,6)
o9 = true

i10 : R = ZZ/101[t]
o10 = R
o10 : PolynomialRing

i11 : R/t === R/t
o11 = false
