Ring extensions

Algebraic structure → Ring theory Ring theory
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In mathematics, a subring of a ring $R$ is a subset of $R$ 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 $R$ .^[a]

Definition

A subring of a ring $(R, +, *, 0, 1)$ is a subset $S$ of $R$ that preserves the structure of the ring, i.e. a ring $(S, +, *, 0, 1)$ with $S \subseteq R$ . Equivalently, it is both a subgroup of $(R, +, 0)$ and a submonoid of $(R, *, 1)$ .

Equivalently, $S$ is a subring if and only if it contains the multiplicative identity of $R$ , and is closed under multiplication and subtraction. This is sometimes known as the subring test.^[1]

Variations

Some mathematicians define rings without requiring the existence of a multiplicative identity (see Ring (mathematics) § History). In this case, a subring of $R$ is a subset of $R$ that is a ring for the operations of $R$ (this does imply it contains the additive identity of $R$ ). 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 $R$ . With the definition requiring a multiplicative identity, which is used in the rest of this article, the only ideal of $R$ that is a subring of $R$ is $R$ itself.

Examples

The ring of integers $\mathbb {Z}$ is a subring of both the field of real numbers and the polynomial ring $\mathbb {Z} [X]$ .^[1]

$\mathbb {Z}$ and its quotients $\mathbb {Z} /n\mathbb {Z}$ have no subrings (with multiplicative identity) other than the full ring.^[1]

Every ring has a unique smallest subring, isomorphic to some ring $\mathbb {Z} /n\mathbb {Z}$ with n a nonnegative integer (see Characteristic). The integers $\mathbb {Z}$ correspond to n = 0 in this statement, since $\mathbb {Z}$ is isomorphic to $\mathbb {Z} /0\mathbb {Z}$ .^[2]

The center of a ring $R$ is a subring of $R$ , and $R$ is an associative algebra over its center.

Subring generated by a set

A special kind of subring of a ring $R$ is the subring generated by a subset $X$ , which is defined as the intersection of all subrings of $R$ containing $X$ .^[3] The subring generated by $X$ is also the set of all linear combinations with integer coefficients of products of elements of $X$ , including the additive identity ("empty combination") and multiplicative identity ("empty product").^[4]

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

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

Ring extension

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

Adjoining

If $A$ is a ring and $T$ is a subring of $A$ generated by $R \cup S$ , where $R$ is a subring, then $T$ is a ring extension and is said to be $S$ adjoined to $R$ , denoted $R [S]$ . Individual elements can also be adjoined to a subring, denoted $R [a 1, a 2, ..., a n]$ .^[5]^[3]

For example, the ring of Gaussian integers $\mathbb {Z} [i]$ is a subring of $\mathbb {C}$ generated by $\mathbb {Z} \cup \{i\}$ , and thus is the adjunction of the imaginary unit $i$ to $\mathbb {Z}$ .^[3]

Prime subring

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

The prime subring of a ring $R$ is a subring of the center of $R$ , which is isomorphic either to the ring $\mathbb {Z}$ of the integers or to the ring of the integers modulo $n$ , where $n$ is the smallest positive integer such that the sum of $n$ copies of $1$ equals $0$ .

Notes

In general, not all subsets of a ring $R$ are rings.
Not to be confused with the ring-theoretic analog of a group extension.

References

Dummit, David Steven; Foote, Richard Martin (2004). Abstract algebra (Third ed.). Hoboken, NJ: John Wiley & Sons. p. 228. ISBN 0-471-43334-9.
Lang, Serge (2002). Algebra (3 ed.). New York. pp. 89–90. ISBN 978-0387953854.{{cite book}}: CS1 maint: location missing publisher (link)
Lovett, Stephen (2015). "Rings". Abstract Algebra: Structures and Applications. Boca Raton: CRC Press. pp. 216–217. ISBN 9781482248913.
Robinson, Derek J. S. (2022). Abstract Algebra: An Introduction with Applications (3rd ed.). Walter de Gruyter GmbH & Co KG. p. 109. ISBN 9783110691160.
Gouvêa, Fernando Q. (2012). "Rings and Modules". A Guide to Groups, Rings, and Fields. Washington, DC: Mathematical Association of America. p. 145. ISBN 9780883853559.

General references

Adamson, Iain T. (1972). Elementary rings and modules. University Mathematical Texts. Oliver and Boyd. pp. 14–16. ISBN 0-05-002192-3.
Sharpe, David (1987). Rings and factorization. Cambridge University Press. pp. 15–17. ISBN 0-521-33718-6.