Irreducibles (Ring theory )

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?)

im struggling on the meaning of irreducible, so if anyone can shed some
light on this, it would be greatly appreciated

Po



.

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