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Pick out the true statement(s):

(a) The set of all $2\times 2$ matrices with rational entries (with the usual operations of matrix addition and matrix multiplication) is a ring which has no nontrivial ideals.

(b) Let $R = C[0, 1]$ be considered as a ring with the usual operations of pointwise addition and pointwise multiplication. Let $I = \{f : [0, 1] → R \mid f(1/2) = 0\}$. 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.

obviously (c) is true but no idea about (a) & (b)

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    I think you're using $\,R\,$ in (b) with two different meanings. Perhaps you meant $\,\Bbb R\,$ for the $\,R\,$ in the definition of $\,I\,$ ...?2012-09-17

4 Answers 4