Answer to Question #137502 in Abstract Algebra for Ram

Question #137502
Is the ideal generated by x^2+1 in Z2[x] a prime ideal of Z2[x]? give reason.
1
Expert's answer
2020-10-11T17:52:02-0400

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.


Need a fast expert's response?

Submit order

and get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!

Comments

No comments. Be the first!

Leave a comment

LATEST TUTORIALS
New on Blog
APPROVED BY CLIENTS