An ideal "P" of a commutative ring "R" is prime if it has the following two properties:
1) If "a" and "b" are two elements of "R" such that their product "ab" is an element of "P", then "a" is in "P" or "b" is in "P";
2) "P" is not the whole ring "R".
Let "f(x)=x+1\\in\\mathbb Z_2[x]" and "g(x)=x+1\\in\\mathbb Z_2[x]". Since "1+1=0" in the ring "\\mathbb Z_2", "f(x)g(x)=(x+1)(x+1)=x^2+x+x+1=x^2+(1+1)x+1=x^2+1" is in the ideal "I=\\langle x^2+1\\rangle" generated by "x^2+1." Taking into account that "f(x)\\notin I" and "g(x)\\notin I," we conclude that the ideal "I=\\langle x^2+1\\rangle" generated by "x^2+1" is not prime.
Comments
Leave a comment