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Subrings
Stable for the composition in the Ring R

Let (R, +, . >) be a ring and let S be a non-empty subset of R, i.e. S ⊂ R. Then S is stable for the addition and multiplication composition in R if



Induced composition

Let a non-empty subset S of the ring (R, +, . >) be stable for the addition and multiplication compositions defined in R. Then we say that the compositions ‘+’, ‘.‘ in R have induced compositions in S. These compositions in S are called induced compositions.

Subring

Let (R, +, .) be a ring, then a non-empty subset S ⊂ R is called a subring of R, if S is stable for the compositions in R and S itself is a ring with respect to these induced compositions. In other words, a non-empty subset S of a ring (R, +, .) if
   
(i) ∀ a, b S a + b S and ab S.
   
(ii) (S, +, .) forms a ring.

Examples of subrings
   
1. The ring of even integers is a subring of the ring of integers (Z, +, .).
   
2. The ring of rational numbers is a subring of the ring of real numbers (R, +, .).
   
3. The ring of Gaussian integers is a subring of the ring of complex numbers (C, +, .).
   
4. The set of all real valued differentiable functions on [0, 1] is a subring of the ring of all real-valued continuous functions.
   
5. The set mZ of multiples of integers by a non-zero number m, m ≠ 0 is a subring of (Z, +, .).
   
6. The ring of integers is a subring of the ring of rational numbers (Q, +, .).
   


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