I'm unable to think of an example of a maximal ideal that is not prime in a finite commutative ring without unity.(adsbygoogle = window.adsbygoogle || []).push({});

Of course the context is, the result stating that in a commutative ring with unity, an ideal is maximal iff it is prime.

Somebody help me out with this ?

Another question is for the ring of 2 x 2 matrices over reals (lets call it M), its subring of matrices with all entries equal (lets call it R) is a ring with a different unity. (wrt usual matrix operations of addition and multiplication)

the unity is 1/2 in each entry.

Of course all non-zero elements of R are invertible, the inverse being (1/4)x (where x is the 'equal entry' for the element of R).

My question is why does this happen, though such matrices belonging to R are all singular.

My own reasoning is that since the determinant is 0 for all matrices belonging to R, we are really not defining anything by the determinant calculation. Also the adjoint of any element of R doesn't belong to R in this case, since the entries will not be equal for adjoint (negatives for the (2,1)th and (1,2)th entry).

But, I'm really not convinced by my own reasons. I know it's tantalizing. Does it have to with the necessary and sufficient criterion relating the determinant with the fact of invertibility of the matrix?

Help me out with this too, someone?

Thank you.

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# Maximal ideal question, + invertibility

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