Description
i1 : R = QQ[a,b,c]
o1 = R
o1 : PolynomialRing

i2 : I = ideal vars R
o2 = ideal (a, b, c)
o2 : Ideal of R

i3 : M = I / I^2
o3 = subquotient ( a b c ,  a2 ab ac b2 bc c2 )
1
o3 : Rmodule, subquotient of R

There is a difference between typing I/J and (I+J)/J in Macaulay2, although conceptually they are the same module. The former has as its generating set the generators of I, while the latter has as its (redundant) generators the generators of I and J. Generally, the former method is preferable.
i4 : gens M
o4 =  a b c 
1 3
o4 : Matrix R < R

i5 : N = (I + I^2)/I^2
o5 = subquotient ( a b c a2 ab ac b2 bc c2 ,  a2 ab ac b2 bc c2 )
1
o5 : Rmodule, subquotient of R

i6 : gens N
o6 =  a b c a2 ab ac b2 bc c2 
1 9
o6 : Matrix R < R
