Some constructions of rings D. J. Bernstein 19980121

Given  construct  Examples
prime ideal P of ring R  R/P, the quotient of R by P  Z[x]/(x^{2} + 3); C[x, y]/(y^{2} − x^{3} − 1)
ideal I of ring R  R/I, the quotient of R by I  Z/10; R[x]/(x^{4} − 1); C[x]/x^{3}
rings A, B  A × B, the direct sum of A and B
rings A, B  A ⊗_{Z} B, the tensor product of A and B over Z
ring R; R-algebras A, B  A ⊗_{R} B, the tensor product of A and B over R
ring R R[x], the polynomial ring over R C[x]; (Z/2)[x]; Z[x]; C[x, y]
subring R of ring S; α ∈ S  R[α], the smallest ring containing R ∪ \{α\} Z[i]; Z[\sqrt{−3}]; Z[i, \sqrt{−3}]
subring R of ring S the integral closure of R in S Z[i]; Z[(1 + \sqrt{−3})/2]
subfield K of field L; α ∈ L K(α), the smallest field containing K ∪ \{α\} Q(i); Q(\sqrt{−3}); Q(i, \sqrt{−3})
ring R  R[[x]], the power series ring over R  C[[x]]; (Z/2)[[x]]; Z[[x]]; C[[x, y]]
prime ideal P of ring R  the completion of R at P  Z_{2}; C[[x]]
ideal I of ring R  the completion of R at I  Z_{10}
prime ideal P of ring R  (R − P)^{−1}R, the localization of R at P  Q; (1 + 2Z)^{−1}Z; C(x)
ring R; multiplicative U ⊆ R U^{−1}R  (1/2)^{N}Z

Note that Z_{2} does not mean Z/2; it means the ring of 2-adic integers.