* Nilradical of a Lie algebra, the nilradical of a Lie algebra is a nilpotent ideal, which is as large as possible

For purposes of comparison, consider the nilradical of a commutative ring, which consists of all elements which are nilpotent.

These notions are of course imprecise, but at least explain why the nilradical of a commutative ring is contained in the ring's Jacobson radical.

If N is the nilradical of commutative ring R, then the quotient ring R / N has no nilpotent elements.

In algebra, the nilradical of a commutative ring is the ideal consisting of the nilpotent elements of the ring.

The nilradical of a commutative ring is the set of all nilpotent elements in the ring, or equivalently the radical of the zero ideal.

If the ring is artinian, the nilradical is its maximal nilpotent ideal.

If the ring R is a finitely generated Z-algebra, then the nilradical is equal to the Jacobson radical, and more generally: the radical of any ideal I will always be equal to the intersection of all the maximal ideals of R that contain I.

The sum of the nil ideals of a ring R is the upper nilradical Nil < sup >*</ sup > R or Köthe radical and is the unique largest nil ideal of R. Köthe's conjecture asks whether any left nil ideal is in the nilradical.

* A / nil ( A ) is a semisimple ring, where nil ( A ) is the nilradical of A.

The nilpotent elements of a commutative ring A form an ideal of A, the so-called nilradical of A ; therefore a commutative ring is reduced if and only if its nilradical is reduced to zero.

* If A is a commutative ring and N is the nilradical of A, then the quotient ring A / N is reduced.

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 ).

A ring homomorphism of commutative rings determines a morphism of Kähler differentials which sends an element dr to d ( f ( r )), the exterior differential of f ( r ).

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.

Our new large-package ring twister for glass fiber yarns is performing well in our customers' mills.

If you walk into the ring because it is fun to show your dog, he will feel it and give you a good performance!!

`` The reason you are in the ring today is to show your ability to present to any judge the most attractive picture of your dog that the skillful use of your aids can produce.

This is the tale of one John Enright, an American who has accidentally killed a man in the prize ring and is now trying to forget about it in a quiet place where he may become a quiet man.

`` He has married me with a ring of bright water '', begins the Kathleen Raine poem from which Maxwell takes his title, and it is this mystic bond between the human and natural world that the author conveys.

Many theorems which are provable using choice are of an elegant general character: every ideal in a ring is contained in a maximal ideal, every vector space has a basis, and every product of compact spaces is compact.

The field F is algebraically closed if and only if the only irreducible polynomials in the polynomial ring F are those of degree one.

The ceremony of such a blessing is similar in some aspects to the consecration of a bishop, with the new abbot being presented with the mitre, the ring, and the crosier as symbols of office and receiving the laying on of hands and blessing from the celebrant.

The sum, difference, product and quotient of two algebraic numbers is again algebraic ( this fact can be demonstrated using the resultant ), and the algebraic numbers therefore form a field, sometimes denoted by A ( which may also denote the adele ring ) or < span style =" text-decoration: overline ;"> Q </ span >.

If K is a number field, its ring of integers is the subring of algebraic integers in K, and is frequently denoted as O < sub > K </ sub >.

In the context of abstract algebra, for example, a mathematical object is an algebraic structure such as a group, ring, or vector space.

Generally speaking, negation is an automorphism of any abelian group, but not of a ring or field.

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

The same definition holds in any unital ring or algebra where a is any invertible element.

The configuration of six carbon atoms in aromatic compounds is known as a benzene ring, after the simplest possible such hydrocarbon, benzene.

The structure is also illustrated as a circle around the inside of the ring to show six electrons floating around in delocalized molecular orbitals the size of the ring itself.

In aromatic substitution one substituent on the arene ring, usually hydrogen, is replaced by another substituent.

When there is more than one substituent present on the ring, their spatial relationship becomes important for which the arene substitution patterns ortho, meta, and para are devised.

This is seen in, for example, phenol ( C < sub > 6 </ sub > H < sub > 5 </ sub >- OH ), which is acidic at the hydroxyl ( OH ), since a charge on this oxygen ( alkoxide-O < sup >–</ sup >) is partially delocalized into the benzene ring.