Finding the ideals of an algebra

[DeldotB]

Say I have the algebra $\mathbb{Z}_5[x]/I$ where is the $I$ is the principle ideal generated by $x^2+4$. How do I find the ideals in A? I can't seem to find an explanation that is clear anywhere. Thanks!

 

[DeldotB]

anyone?

 

[HallsofIvy]

I am a bit confused by your "How do I find the ideals in A?" when there is no mention of A! Did you mean $A= Z_5 [x]/I$?

A subset, S, of an algebra, A, is an "ideal" if and only if the product of a member of s with any member of A is again in S. $Z_5[x]$ is the set of all polynomials of degree 5 or less with integer coefficients. I is the set of all such polynomials of the form $(x^2+ 4)Z_3[x]$ where $Z_3[x]$ is the set of all polynomials of degree 3 or less with integer coefficients.