prime subring

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English

Noun

prime subring (plural prime subrings)

(algebra, ring theory, of a (unital) ring $R$ ) The image of the natural homomorphism from $\mathbb {Z}$ $\mathbb {Z}$ into $R$ $R$ , which is the unique homomorphism mapping $1_{\mathbb {Z} }$ $1_{\mathbb {Z} }$ to $1_{R}$ $1_{R}$ . Equivalently, the intersection of all subrings of $R$ $R$ .
Synonym: prime ring
- 2006, Shreeram Shankar Abhyankar, The Algebraic Structure of Crossed Products, World Scientific, page 21:
  The image of the homomorphism ℤ→ R is called the prime subring of R. Clearly ch(R) = 0 ⟷ the prime subring of R is isomorphic to ℤ.
- 2013, Michel Broué, Some Topics in Algebra: An Advanced Undergraduate Course at PKU, Springer Science & Business Media, page 10:
  The image of the morphism f_A is the prime subring of A. The prime subring is either isomorphic to the ring ℤ or isomorphic to the ring ℤ/nℤ for some n > 0.

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