Ring extension

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In mathematics, more specifically in ring theory, a ring extension or extension ring is a ring R with a subring S. We write R/S and say R is a ring extension of S

Given an extension R/S of commutative rings and a prime ideal P of R, it follows that the intersection, say p, of P with S is a prime ideal of S. In this case we say that P lies over p. The situation is more complicated when R is not commutative.

Examples[edit]

  • A field extension is a special case of ring extension. Polynomial ring R[x] is an extension of R.

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