Subring Explained

In mathematics, a subring of a ring is a subset of that is itself a ring when binary operations of addition and multiplication on R are restricted to the subset, and that shares the same multiplicative identity as .^[1]

Definition

A subring of a ring is a subset of that preserves the structure of the ring, i.e. a ring with . Equivalently, it is both a subgroup of and a submonoid of .

Equivalently, is a subring if and only if it contains the multiplicative identity of, and is closed under multiplication and subtraction. This is sometimes known as the subring test.^[2]

Variations

Some mathematicians define rings without requiring the existence of a multiplicative identity (see ). In this case, a subring of is a subset of that is a ring for the operations of (this does imply it contains the additive identity of). This alternate definition gives a strictly weaker condition, even for rings that do have a multiplicative identity, in that all ideals become subrings, and they may have a multiplicative identity that differs from the one of . With the definition requiring a multiplicative identity, which is used in the rest of this article, the only ideal of that is a subring of is itself.

Examples

\Z is a subring of both the field of real numbers and the polynomial ring \Z[X] .Z and its quotients Z /n Z have no subrings (with multiplicative identity) other than the full ring.

• Every ring has a unique smallest subring, isomorphic to some ring

Z /n Z with n a nonnegative integer (see Characteristic). The integers Z correspond to in this statement, since Z is isomorphic to Z / 0 Z .^[3]

• The center of a ring is a subring of, and is an associative algebra over its center.
• The ring of split-quaternions has subrings isomorphic to the rings of dual numbers and split-complex numbers, and to the complex number field.

Subring generated by a set

See also: Generator (mathematics).

A special kind of subring of a ring is the subring generated by a subset, which is defined as the intersection of all subrings of containing .^[4] The subring generated by is also the set of all linear combinations with integer coefficients of elements of, including the additive identity ("empty combination") and multiplicative identity ("empty product").

Any intersection of subrings of is itself a subring of ; therefore, the subring generated by (denoted here as) is indeed a subring of . This subring is the smallest subring of containing ; that is, if is any other subring of containing, then .

Since itself is a subring of, if is generated by, it is said that the ring is generated by .

Ring extension

Subrings generalize some aspects of field extensions. If is a subring of a ring, then equivalently is said to be a ring extension^[5] of .

Adjoining

If is a ring and is a subring of generated by, where is a subring, then is a ring extension and is said to be adjoined to, denoted . Individual elements can also be adjoined to a subring, denoted .^[6]

\Z[i] is a subring of \C generated by \Z \cup \{i\} , and thus is the adjunction of the imaginary unit to \Z .

Prime subring

The intersection of all subrings of a ring is a subring that may be called the prime subring of by analogy with prime fields.

The prime subring of a ring is a subring of the center of, which is isomorphic either to the ring

\Z of the integers or to the ring of the integers modulo , where is the smallest positive integer such that the sum of copies of equals .

See also

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

References

General references

• Book: Adamson . Iain T. . Elementary rings and modules . University Mathematical Texts . Oliver and Boyd . 1972 . 0-05-002192-3 . 14–16.
• Book: Sharpe . David . Rings and factorization . registration . . 1987 . 0-521-33718-6 . 15–17.

Notes and References

1. In general, not all subsets of a ring are rings.
2. Book: Dummit . David Steven . Foote . Richard Martin . Abstract algebra . 2004 . John Wiley & Sons . Hoboken, NJ . 0-471-43334-9 . Third . 228.
3. Book: Lang . Serge . Algebra . 2002 . New York . 978-0387953854 . 3 . 89-90.
4. Book: Lovett, Stephen . 2015 . Abstract Algebra: Structures and Applications . Rings . 216-217 . CRC Press . Boca Raton . 9781482248913.
5. Not to be confused with the ring-theoretic analog of a group extension.
6. Book: Gouvêa, Fernando Q. . Fernando Q. Gouvêa . 2012 . A Guide to Groups, Rings, and Fields . Rings and Modules . 145 . Mathematical Association of America . Washington, DC . 9780883853559.