How Do You Know Is It a Subring of Something

In mathematics, a subring of R is a subset of a ring that is itself a ring when binary operations of improver and multiplication on R are restricted to the subset, and which shares the aforementioned multiplicative identity equally R. For those who define rings without requiring the being of a multiplicative identity, a subring of R is just a subset of R that is a band for the operations of R (this does imply it contains the additive identity of R). The latter gives a strictly weaker status, even for rings that do take a multiplicative identity, then that for instance all ideals become subrings (and they may have a multiplicative identity that differs from the i of R). With definition requiring a multiplicative identity (which is used in this commodity), the only platonic of R that is a subring of R is R itself.

Definition [edit]

A subring of a band (R, +, ∗, 0, i) is a subset S of R that preserves the structure of the band, i.east. a ring (South, +, ∗, 0, 1) with S ⊆ R . Equivalently, information technology is both a subgroup of (R, +, 0) and a submonoid of (R, ∗, i).

Examples [edit]

The ring ${\displaystyle \mathbb {Z} }$ and its quotients ${\displaystyle \mathbb {Z} /northward\mathbb {Z} }$ have no subrings (with multiplicative identity) other than the full ring.

Every ring has a unique smallest subring, isomorphic to some ring ${\displaystyle \mathbb {Z} /northward\mathbb {Z} }$ with northward a nonnegative integer (see characteristic). The integers ${\displaystyle \mathbb {Z} }$ stand for to north = 0 in this argument, since ${\displaystyle \mathbb {Z} }$ is isomorphic to ${\displaystyle \mathbb {Z} /0\mathbb {Z} }$ .

Subring test [edit]

The subring test is a theorem that states that for whatever band R, a subset South of R is a subring if and simply if information technology is closed under multiplication and subtraction, and contains the multiplicative identity of R.

Every bit an example, the ring Z of integers is a subring of the field of existent numbers and too a subring of the band of polynomials Z[X].

Ring extensions [edit]

Not to exist confused with Band extension, a ring-theoretic analog of a group extension.

If S is a subring of a ring R, then equivalently R is said to be a ring extension of S, written as R/South in similar notation to that for field extensions.

Subring generated past a set [edit]

Let R be a band. Whatever intersection of subrings of R is again a subring of R. Therefore, if 10 is any subset of R, the intersection of all subrings of R containing X is a subring S of R. South is the smallest subring of R containing X. ("Smallest" ways that if T is any other subring of R containing X, then Due south is contained in T.) South is said to be the subring of R generated past X. If Due south = R, we may say that the ring R is generated past Ten.

Relation to ideals [edit]

Proper ideals are subrings (without unity) that are closed nether both left and right multiplication by elements of R.

If 1 omits the requirement that rings have a unity element, and then subrings need only exist not-empty and otherwise adapt to the ring structure, and ethics become subrings. Ethics may or may not have their own multiplicative identity (singled-out from the identity of the ring):

• The platonic I = {(z,0) | z in Z} of the ring Z × Z = {(ten,y) | x,y in Z} with componentwise addition and multiplication has the identity (ane,0), which is dissimilar from the identity (1,ane) of the ring. So I is a band with unity, and a "subring-without-unity", but not a "subring-with-unity" of Z × Z.
• The proper ideals of Z accept no multiplicative identity.

If I is a prime number ideal of a commutative ring R, then the intersection of I with any subring S of R remains prime number in S. In this case one says that I lies over I ∩South. The situation is more complicated when R is not commutative.

Profile past commutative subrings [edit]

A ring may be profiled^{[ clarification needed ]} by the diversity of commutative subrings that it hosts:

• The quaternion ring H contains only the complex aeroplane equally a planar subring
• The coquaternion ring contains three types of commutative planar subrings: the dual number plane, the split-complex number plane, too equally the ordinary complex plane
• The band of 3 × iii real matrices also contains 3-dimensional commutative subrings generated by the identity matrix and a nilpotent ε of guild 3 (εεε = 0 ≠ εε). For instance, the Heisenberg group tin can be realized as the join of the groups of units of two of these nilpotent-generated subrings of 3 × 3 matrices.

See too [edit]

• Integral extension
• Group extension
• Algebraic extension
• Ore extension

References [edit]

• Iain T. Adamson (1972). Elementary rings and modules. University Mathematical Texts. Oliver and Boyd. pp. xiv–16. ISBN0-05-002192-3.
• Page 84 of Lang, Serge (1993), Algebra (Third ed.), Reading, Mass.: Addison-Wesley, ISBN978-0-201-55540-0, Zbl 0848.13001
• David Sharpe (1987). Rings and factorization . Cambridge University Press. pp. 15–17. ISBN0-521-33718-six.

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Source: https://en.wikipedia.org/wiki/Subring