( The definition of a homomorphism depends on the type of algebraic structure ; see, for example: group homomorphism, ring homomorphism, and linear operator ).

* A field automorphism is a bijective ring homomorphism from a field to itself.

Starting with a ring A, we get a unital associative R-algebra by providing a ring homomorphism whose image lies in the center of A.

A homomorphism between two associative R-algebras is an R-linear ring homomorphism.

The unique ring homomorphism from Z to A is determined by the fact that it must send 1 to the identity in A.

For the case of a non-commutative base ring R and a right module M < sub > R </ sub > and a left module < sub > R </ sub > N, we can define a bilinear map, where T is an abelian group, such that for any n in N, is a group homomorphism, and for any m in M, is a group homomorphism too, and which also satisfies

* A ring homomorphism is a homomorphism between two rings.

For example, a ring possesses both addition and multiplication, and a homomorphism from the ring to the ring is a function such that

Even though the set may be the same, the same function might be a homomorphism, say, in group theory ( sets with a single operation ) but not in ring theory ( sets with two related operations ), because it fails to preserve the additional operation that ring theory considers.

However, the same is not true for epimorphisms ; for instance, the inclusion of Z as a ( unitary ) subring of Q is not surjective, but an epimorphic ring homomorphism.

This inclusion thus also is an example of a ring homomorphism which is both mono and epi, but not iso.

If a is an idempotent of the endomorphism ring End < sub > R </ sub >( M ), then the endomorphism is an R module involution of M. That is, f is an R homomorphism such that f < sup > 2 </ sup > is the identity endomorphism of M.

Thus the set of all polynomials with coefficients in the ring R forms itself a ring, the ring of polynomials over R, which is denoted by R. The map from R to R sending r to rX < sup > 0 </ sup > is an injective homomorphism of rings, by which R is viewed as a subring of R. If R is commutative, then R is an algebra over R.

In mathematics, an associative algebra A is an associative ring that has a compatible structure of a vector space over a certain field K or, more generally, of a module over a commutative ring R. Thus A is endowed with binary operations of addition and multiplication satisfying a number of axioms, including associativity of multiplication and distributivity, as well as compatible multiplication by the elements of the field K or the ring R.

Let R be a fixed commutative ring.

If A itself is commutative ( as a ring ) then it is called a commutative R-algebra.

* Any commutative ring R is an algebra over itself, or any subring of R.

* Any ring of matrices with coefficients in a commutative ring R forms an R-algebra under matrix addition and multiplication.

* Every polynomial ring R ..., x < sub > n </ sub > is a commutative R-algebra.

The definition works without any changes if instead of vector spaces over a field F, we use modules over a commutative ring R. It also can be easily generalized to n-ary functions, where the proper term is multilinear.

Let A be a unital commutative Banach algebra over C. Since A is then a commutative ring with unit, every non-invertible element of A belongs to some maximal ideal of A.

( valid for any elements x, y of a commutative ring ),

* The spectrum of any commutative ring with the Zariski topology ( that is, the set of all prime ideals ) is compact, but never Hausdorff ( except in trivial cases ).

Two ideals A and B in the commutative ring R are called coprime ( or comaximal ) if A + B = R. This generalizes Bézout's identity: with this definition, two principal ideals ( a ) and ( b ) in the ring of integers Z are coprime if and only if a and b are coprime.

Although most often used for matrices whose entries are real or complex numbers, the definition of the determinant only involves addition, subtraction and multiplication, and so it can be defined for square matrices with entries taken from any commutative ring.

For square matrices with entries in a non-commutative ring, for instance the quaternions, there is no unique definition for the determinant, and no definition that has all the usual properties of determinants over commutative rings.

Provided the underlying scalars form a field ( more generally, a commutative ring with unity ), the definition below shows that such a function exists, and it can be shown to be unique.

The center of a division ring is commutative and therefore a field.

In abstract algebra, a field is a commutative ring which contains a multiplicative inverse for every nonzero element, equivalently a ring whose nonzero elements form an abelian group under multiplication.

The notion of greatest common divisor can more generally be defined for elements of an arbitrary commutative ring, although in general there need not exist one for every pair of elements.

For bridged alkenes, the Bredt's rule states that a double bond cannot be placed at the bridgehead of a bridged ring system, unless the rings are large enough.

In Thomson's model, electrons were free to rotate in rings which were further stabilized by interactions between the electrons, and spectra were to be accounted for by energy differences of different ring orbits.

Still, Thomson's model ( along with a similar Saturnian ring model for atomic electrons, also put forward in 1904 by Nagaoka after James Clerk Maxwell's model of Saturn's rings ), were earlier harbingers of the later and more successful solar-system-like Bohr model of the atom.

This ring has the property that a · a = a for all a in A ; rings with this property are called Boolean rings.

There are four gearing options: two-speed hub gear integrated with chain ring, up to 3 chain rings, up to 11 sprockets, hub gear built in to rear wheel ( 3-speed to 14-speed ).

Complex-crater morphology on rocky planets appears to follow a regular sequence with increasing size: small complex craters with a central topographic peak are called central peak craters, for example Tycho ; intermediate-sized craters, in which the central peak is replaced by a ring of peaks, are called peak-ring craters, for example Schrödinger ; and the largest craters contain multiple concentric topographic rings, and are called multi-ringed basins, for example Orientale.

Several patterns of linking the rings together have been known since ancient times, with the most common being the 4-to-1 pattern ( where each ring is linked with four others ).

The nucleobases are classified into two types: the purines, A and G, being fused five-and six-membered heterocyclic compounds, and the pyrimidines, the six-membered rings C and T. A fifth pyrimidine nucleobase, uracil ( U ), usually takes the place of thymine in RNA and differs from thymine by lacking a methyl group on its ring.

In Norse mythology, Draupnir ( Old Norse " the dripper ") is a gold ring possessed by the god Odin with the ability to multiply itself: Every ninth night eight new rings ' drip ' from Draupnir, each one of the same size and weight as the original.

: Odin laid upon the pyre the gold ring called Draupnir ; this quality attended it: that every ninth night there fell from it eight gold rings of equal weight.

For example, the proof that the column rank of a matrix over a field equals its row rank yields for matrices over division rings only that the left column rank equals its right row rank: it does not make sense to speak about the rank of a matrix over a division ring.

Typically, elf circles were fairy rings consisting of a ring of small mushrooms, but there was also another kind of elf circle:

If a decomposition exists with each c < sub > i </ sub > a centrally primitive idempotent, then R is a direct sum of the corner rings c < sub > i </ sub > Rc < sub > i </ sub >, each of which is ring irreducible.

By the Chinese Remainder Theorem, this ring factors into the direct product of rings of integers mod p. Now each of these factors is a field, so it is clear that the only idempotents will be 0 and 1.

This same construction may be carried out if the A < sub > i </ sub >' s are sets, rings, modules ( over a fixed ring ), algebras ( over a fixed field ), etc., and the homomorphisms are homomorphisms in the corresponding category.

* The ring of p-adic integers is the inverse limit of the rings Z / p < sup > n </ sup > Z ( see modular arithmetic ) with the index set being the natural numbers with the usual order, and the morphisms being " take remainder ".

* The ring of formal power series over a commutative ring R can be thought of as the inverse limit of the rings, indexed by the natural numbers as usually ordered, with the morphisms from to given by the natural projection.

Luthor is quickly overwhelmed by his greed, and sets out to steal the rings of his fellow inducted Lanterns, taking Scarecrow's yellow ring and attempting to steal Mera's red one, but is held back by the Atom ( wielding the ring-staff of the Indigo tribe ) and the Flash wearing a Blue Lantern Ring.

: The standard signature of rings is σ < sub > ring </ sub > =

* A rigid pentacyclic structure consisting of a benzene ring ( A ), two partially unsaturatedcyclohexane rings ( B and C ), a piperidine ring ( D ) and a tetrahydrofuran ring ( E ).