Re: Irreducibles (Ring theory )

po wrote:
> Im trying to understand what an irreducible element is.
> i have my definition
> an element (r) is irreducible if
> it is not equal to 0
> r is not invertible r=ab a or b is invertible
>
> then in a text book i have the irreducibles of the integers are all
> the prime numbers and their negatives...
> but what about the other numbers? whats the inverse of 4? (in the
> integers, it doesnt exist, so why is this not an irreducible?)

Let's check the criteria. In the ring of integers:

4 is not equal to 0.
4 is not invertible.
4 = 2*2, and 2 is not invertible.

So 4 is not irreducible.

The really important part is that last criterion. It says you can't
produce a "nontrivial factorization" of an irreducible.

.

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