| Revision as of 00:26, 11 April 2010 edit68.112.193.62 (talk) There was an unnecessary comma near the beginning of this definition.← Previous edit | Revision as of 16:10, 23 October 2010 edit undo209.147.105.130 (talk) →Relation to idealsNext edit → | ||
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| ==Relation to ideals== | ==Relation to ideals== | ||
| Proper ]s are |
Proper ]s are subrings that are closed under both left and right multiplication by elements from ''R''. | ||
| If one omits the requirement that rings have a unit element, then subrings need only be non-empty and be closed under subtraction and multiplication, and ideals become subrings. Ideals may or may not have their own multiplicative identity (distinct from the identity of the ring): | |||
| *The ideal ''I'' = {(''z'',0)|''z'' in '''Z'''} of the ring '''Z''' × '''Z''' = {(''x'',''y'')|''x'',''y'' in '''Z'''} with componentwise addition and multiplication has the identity (1,0), which is different from the identity (1,1) of the ring. So ''I'' is a ring with unity, and a "subring-without-unity", but not a "subring-with-unity" of '''Z''' × '''Z'''. | |||
| *The proper ideals of '''Z''' have no multiplicative identity. | |||
| ==Profile by commutative subrings== | ==Profile by commutative subrings== | ||
Revision as of 16:10, 23 October 2010
In mathematics, a subring is a subset of a ring which contains the multiplicative identity and is itself a ring under the same binary operations. Naturally, those authors who do not require rings to contain a multiplicative identity do not require subrings to possess the identity (if it exists). This leads to the added advantage that ideals become subrings (see below).
A subring of a ring (R, +, *) is a subgroup of (R, +) which contains the mutiplicative identity and is closed under multiplication.
For example, the ring Z of integers is a subring of the field of real numbers and also a subring of the ring of polynomials Z.
The ring Z has no subrings (with multiplicative identity) other than itself.
Every ring has a unique smallest subring, isomorphic to either the integers Z or some ring Z/nZ with n a nonnegative integer (see characteristic).
The subring test states that for any ring, a nonempty subset of that ring is itself a ring if it is closed under multiplication and subtraction, and has a multiplicative identity.
Subring generated by a set
Let R be a ring. Any intersection of subrings of R is again a subring of R. Therefore, if X is any subset of R, the intersection of all subrings of R containing X is a subring S of R. S is the smallest subring of R containing X. ("Smallest" means that if T is any other subring of R containing X, then S is contained in T.) S is said to be the subring of R generated by X. If S = R, we may say that the ring R is generated by X.
Relation to ideals
Proper ideals are subrings that are closed under both left and right multiplication by elements from R.
Profile by commutative subrings
A ring may be profiled by the variety of commutative subrings that it hosts:
- The quaternion ring H contains only the complex plane as a planar subring
- The coquaternion ring contains three types of commutative planar subrings: the dual number plane, the split-complex number plane, as well as the ordinary complex plane
- The ring of 3 x 3 real matrices also contains 3-dimensional commutative subrings generated by the identity matrix and a nilpotent ε of order 3 (εεε = 0 ≠ εε). For instance, the Heisenberg group can be realized as the join of the groups of units of two of these nilpotent-generated subrings of 3 x 3 matrices.
References
- Iain T. Adamson (1972). Elementary rings and modules. University Mathematical Texts. Oliver and Boyd. pp. 14–16. ISBN 0-05-002192-3.
- Page 84 of Lang, Serge (1993), Algebra (Third ed.), Reading, Mass.: Addison-Wesley, ISBN 978-0-201-55540-0, Zbl 0848.13001
- David Sharpe (1987). Rings and factorization. Cambridge University Press. pp. 15–17. ISBN 0-521-33718-6.