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### Quotient of polynomial ring over integers not working

I have a problem. I want Sage to calculate in $\mathbb Z[x]/\langle4,2x,x^2\rangle$ but none of the relations are being calculated/recognized properly:

sage: R.<x> = ZZ[]
sage: I = R.ideal(4,2*x,x^2)
sage: S.<a> = R.quotient(I)
sage: a^2
a^2
sage: 2*a
2*a
sage: S(2)+S(2)
4


When I introduce a superfluous variable $y$ and consider $\mathbb Z[x,y]/\langle4,2x,x^2,y\rangle$, which is practically the same ring (i.e. isomorphic), then it seems to work:

sage: R.<x,y> = ZZ[]
sage: I = R.ideal(4,2*x,x^2,y)
sage: S.<a,b> = R.quotient(I)
sage: a^2
0
sage: 2*a
0
sage: S(2)+S(2)
0


But now look at this:

sage: S(2)+S(3)
5


How can I solve this problem?

### Quotient of polynomial ring over integers not working

I have a problem. I want Sage to calculate in $\mathbb Z[x]/\langle4,2x,x^2\rangle$ but none of the relations are being calculated/recognized properly:

sage: R.<x> = ZZ[]
sage: I = R.ideal(4,2*x,x^2)
sage: S.<a> = R.quotient(I)
sage: a^2
a^2    # the output should be 0
a^2
sage: 2*a
2*a    # the output should be 0
2*a
sage: S(2)+S(2)
S(2)+S(2)    # the output should be 0
4


When I introduce a superfluous variable $y$ and consider $\mathbb Z[x,y]/\langle4,2x,x^2,y\rangle$, which is practically the same ring (i.e. isomorphic), then it seems to work:

sage: R.<x,y> = ZZ[]
sage: I = R.ideal(4,2*x,x^2,y)
sage: S.<a,b> = R.quotient(I)
sage: a^2
0
sage: 2*a
0
sage: S(2)+S(2)
0


But now look at this:

sage: S(2)+S(3)
S(2)+S(3)    # the output should be 1
5


How can I solve this problem?

### Quotient of polynomial ring over integers not working

I have a problem. I want Sage to calculate in $\mathbb Z[x]/\langle4,2x,x^2\rangle$ but none of the relations are being calculated/recognized properly:

sage: R.<x> = ZZ[]
sage: I = R.ideal(4,2*x,x^2)
sage: S.<a> = R.quotient(I)
sage: a^2    # the output should be 0
a^2
sage: 2*a    # the output should be 0
2*a
sage: S(2)+S(2)    # the output should be 0
4


When I introduce a superfluous variable $y$ and consider $\mathbb Z[x,y]/\langle4,2x,x^2,y\rangle$, Z[x,y]/\langle4,2x,x^2,y\rangle$instead, which is practically the same ring (i.e. isomorphic), then it seems to work: sage: R.<x,y> = ZZ[] sage: I = R.ideal(4,2*x,x^2,y) sage: S.<a,b> = R.quotient(I) sage: a^2 0 sage: 2*a 0 sage: S(2)+S(2) 0  But now look at this: sage: S(2)+S(3) # the output should be 1 5  How can I solve this problem?  4 retagged tmonteil 27343 ●31 ●203 ●515 http://wiki.sagemath.o... ### Quotient of polynomial ring over integers not working I have a problem. I want Sage to calculate in$\mathbb Z[x]/\langle4,2x,x^2\rangle$but none of the relations are being calculated/recognized properly: sage: R.<x> = ZZ[] sage: I = R.ideal(4,2*x,x^2) sage: S.<a> = R.quotient(I) sage: a^2 # the output should be 0 a^2 sage: 2*a # the output should be 0 2*a sage: S(2)+S(2) # the output should be 0 4  When I introduce a superfluous variable$y$and consider$\mathbb Z[x,y]/\langle4,2x,x^2,y\rangle$instead, which is practically the same ring (i.e. isomorphic), then it seems to work: sage: R.<x,y> = ZZ[] sage: I = R.ideal(4,2*x,x^2,y) sage: S.<a,b> = R.quotient(I) sage: a^2 0 sage: 2*a 0 sage: S(2)+S(2) 0  But now look at this: sage: S(2)+S(3) # the output should be 1 5  How can I solve this problem?  5 retagged tmonteil 27343 ●31 ●203 ●515 http://wiki.sagemath.o... ### Quotient of polynomial ring over integers not working I have a problem. I want Sage to calculate in$\mathbb Z[x]/\langle4,2x,x^2\rangle$but none of the relations are being calculated/recognized properly: sage: R.<x> = ZZ[] sage: I = R.ideal(4,2*x,x^2) sage: S.<a> = R.quotient(I) sage: a^2 # the output should be 0 a^2 sage: 2*a # the output should be 0 2*a sage: S(2)+S(2) # the output should be 0 4  When I introduce a superfluous variable$y$and consider$\mathbb Z[x,y]/\langle4,2x,x^2,y\rangle\$ instead, which is practically the same ring (i.e. isomorphic), then it seems to work:

sage: R.<x,y> = ZZ[]
sage: I = R.ideal(4,2*x,x^2,y)
sage: S.<a,b> = R.quotient(I)
sage: a^2
0
sage: 2*a
0
sage: S(2)+S(2)
0


But now look at this:

sage: S(2)+S(3)    # the output should be 1
5


How can I solve this problem?