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Pick out the true statement(s):
(a) The set of all $2×2$ matrices with rational entries (with the usual operations of matrix addition and matrix multiplication) is a ring which has no non-trivial ideals.
(b) Let $R = C [0, 1$] be considered as a ring with the usual operations of pointwise addition and pointwise multiplication. Let
I = {$f : [0, 1] → \mathbb{R} | f (1/2) = 0$}. Then $I$ is a maximal ideal.
(c) Let $R$ be a commutative ring and let $P$ be a prime ideal of $R$. Then $R/P$ is an integral domain.

I know that (c) is true but not sure about the others.

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all three are true for a> look at ideals of $\mathbb Q$ and what is its relation with ideals of the matrix ring with rational entries. –  jim Jan 1 '13 at 4:36

1 Answer 1

(a) What are the left ideals of this ring? The right ideals? Hint: Use the left action of this ring on $\mathbb{Q}^2$ to describe the left ideals, and the right action of the ring (via the transpose) to describe the right ideals.

(b) Show that I is an ideal. Then it is a kernel of some homomorphism. What is its image?

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b is ok.thanks. –  gomti Jan 1 '13 at 4:44
but would you explain (a) please. –  gomti Jan 1 '13 at 4:45
This is the ring of linear maps $\mathbb{Q}^2\to \mathbb{Q}^2$. The nontrivial left ideals are the annihilators of lines. –  Brett Frankel Jan 1 '13 at 4:51

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