Exploring Cosets of R=Z_4[x]/((x^2+1)*Z_4[x])

[Stephen88]

I'm trying to list the cosets of the following ring and describe the relations that hold between these cosets.
R=Z_4[x]/((x^2+1)*Z_4[x])
I'm using the division algorithm since x^2+1 is monic in the ring Z_4[x].Now for every f that belongs to Z_4[x] by the division algorithm
f=(x^2+1)q(x)+p(x)=>the cosets are of the the form...a*x+b+I where I is an ideal generated by x^2+1.
x^2+1=0 in the quotient=>a new ring where multiplication between cosets A+I and B+I is is defined by (A+I)(B+I)=(AB)+1 where the relation x^2+1=0 exists
Is is ok?

 

[lippyka]

Yes, that is correct. The cosets of R are of the form a*x+b+I, where I is the ideal generated by x2+1, and the relation between two cosets A+I and B+I is (A+I)(B+I)=(AB)+I, where the relation x2+1=0 holds in the quotient.