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Let R be a ring (with 1). A subset S of R is a subring of R if it is itself a ring with the operations and identity of R.

Equivalently, a subset S of R is a subring if and only if

  • 1R in S
  • a-b in S, for all a,b in S
  • ab in S, for all a,b in S

For example, the ring of integers is a subring of the ring of rational numbers. The collection of upper triangular 2x2 matrices with complex coefficients is a subring of the ring of 2x2 complex matrices.

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