Associative rings and algebras Polynomial identity rings Concept
Polynomial identity ring
Summary
A polynomial identity ring, or PI ring, is a ring in which some nonzero noncommutative polynomial vanishes for all choices of elements. Commutative rings are PI rings, because vanishes on them, and so are matrix rings over commutative rings by the Amitsur–Levitzki theorem. PI rings form the class of noncommutative rings that are closest to commutative rings, and many results of commutative algebra extend to them. Their theory is built on free algebras, central polynomials and generic matrices.
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About this page
| Tag | C0EDT81EGH |
|---|---|
| URI | m2://C0EDT81EGH |
| Kind | concept |
| Writing | W0 stub |
| MSC | 16R10, 16R20 |
| Specializes | Ring |
| Same as |
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Used by
- Central polynomial
- Kaplansky's theorem on PI rings
- Generic matrix algebra
- Kurosh problem
- Posner's theorem
- Amitsur–Levitzki theorem