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Subrings

Stable for the composition in the Ring

Let

Let a non-empty subset

Let

**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, +, .)**.**Services: -**Subrings Homework | Subrings Homework Help | Subrings Homework Help Services | Live Subrings Homework Help | Subrings Homework Tutors | Online Subrings Homework Help | Subrings Tutors | Online Subrings Tutors | Subrings Homework Services | SubringsSubmit Your Query ???

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