An irreducible element (indecomposable element) is one of the basic concepts of ring theory.
Let R be a region of integrity , i.e. commutative ring without zero divisors . An element p ≠ 0 is called irreducible if it is not invertible and from the equality p = bc it follows that either b or c is invertible.
If p ≠ 0 is a simple element , i.e. (p) is a simple ideal , then p is irreducible. In fact, then if p = ab , by virtue of simplicity (p) we have, for example . Then we have: a = px for some x , so a = abx and bx = 1 , i.e. b is reversible. The converse is not true in the general case, although it holds for every factorial ring .
Polynomials over the ring R are called irreducible if they are irreducible elements .
![R [x_ {1}, \ ldots, x_ {n}]](https://wikimedia.org/api/rest_v1/media/math/render/svg/58c388e003e234e12fb55533e35a211c8cf295e5)
Literature
- Van der Waerden B.L. Algebra-M :, Science, 1975
- Zarissky O., Samuel P. Commutative Algebra Vol. 1-M :, IL, 1963
- Leng S. Algebra-M :, World, 1967
See also
- Eisenstein criterion
- Irreducible polynomial