Starting with an R-module A, we get an associative R-algebra by equipping A with an R-bilinear mapping A × A → A such that

This R-bilinear mapping then gives A the structure of a ring and an associative R-algebra.

Every associative R-algebra arises this way.

This definition is equivalent to the statement that a unital associative R-algebra is a monoid in R-Mod ( the monoidal category of R-modules ).

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.

If A is a monoid under A-multiplication ( it satisfies associativity and it has an identity ), then the R-algebra is called an associative algebra.

An equivalent definition of an associative R-algebra is a ring homomorphism such that the image of f is contained in the center of A.

* Associativity: composition of morphisms is always associative.

In mathematics, the associative property is a property of some binary operations.

Within an expression containing two or more occurrences in a row of the same associative operator, the order in which the operations are performed does not matter as long as the sequence of the operands is not changed.

Since this holds true when performing addition on any real numbers, we say that " addition of real numbers is an associative operation.

Formally, a binary operation on a set S is called associative if it satisfies the associative law:

The evaluation order does not affect the value of such expressions, and it can be shown that the same holds for expressions containing any number of operations .< ref > Thus, when is associative, the evaluation order can be left unspecified without causing ambiguity, by omitting the parentheses and writing simply:

The two methods produce the same result ; string concatenation is associative ( but not commutative ).

* Addition and multiplication of complex numbers and quaternions is associative.

Addition of octonions is also associative, but multiplication of octonions is non-associative.

* If M is some set and S denotes the set of all functions from M to M, then the operation of functional composition on S is associative:

In short, composition of maps is always associative.

Premultiplied alpha has some practical advantages over normal alpha blending because premultiplied alpha blending is associative and linear interpolation gives better results, although premultiplication can cause a loss of precision and, in extreme cases, a noticeable loss of quality.

However, this operation may not be appropriate for all applications, since it is not associative.

The associative version of this operation is very similar ; simply take the newly computed color value and divide it by its new alpha value, as follows:

The second assumption is that the operator must respect the associative rule:

The term is also used, especially in the description of algorithms, to mean associative array or " abstract array ", a theoretical computer science model ( an abstract data type or ADT ) intended to capture the essential properties of arrays.

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.

* A supercommutative algebra ( sometimes called a skew-commutative associative ring ) is the same thing as an anticommutative ( Z / 2Z, ε )-graded algebra, where ε is the identity endomorphism for the additive structure.

Addition is typically assumed to be commutative and associative, and the additive identity satisfies 0x = 0 for all x.

If + is associative ( for all,, ), then an additive inverse is unique

For example, since addition of real numbers is associative, each real number has a unique additive inverse.

< li >+ is the additive operation and is a commutative ( i. e., +( a, b ) = +( b, a )) and associative ( i. e., +( a ,+( b, c ))

In fuzzy logic, he introduced fuzzy cognitive maps, fuzzy subsethood, additive fuzzy systems, fuzzy approximation theorems, optimal fuzzy rules, fuzzy associative memories, various neural-based adaptive fuzzy systems, ratio measures of fuzziness, the shape of fuzzy sets, the conditional variance of fuzzy systems, and the geometric view of ( finite ) fuzzy sets as points in hypercubes and its relationship to the on-going debate of fuzziness versus probability.

Briefly, a ring is an abelian group with an additional binary operation that is associative and is distributive over the abelian group operation.

Medial magmas need not be associative: for any nontrivial abelian group and integers, replacing the group operation with the binary operation yields a medial magma which in general is neither associative nor commutative.

This loop of units in an alternative ring or algebra is analogous to the group of units in an associative ring or algebra.

Proof: Notice that the nim-sum (⊕) obeys the usual associative and commutative laws of addition (+), and also satisfies an additional property, x ⊕ x = 0 ( technically speaking, the nonnegative integers under ⊕ form an Abelian group of exponent 2 ).

GALA Choruses, an associative group, now has well-over 100 member choruses throughout the world.

Owing to the above properties ( along with the associative property, which rotations obey ,) the set of all rotations is a group under composition.

Similarly, we can define a weight space V < sub > λ </ sub > for any representation of a Lie group or an associative algebra.

Similarly we can define a highest-weight module for representation of a Lie group or an associative algebra.

Let be the Cartan matrix of the Kac-Moody algebra, and let q be a nonzero complex number distinct from 1, then the quantum group, U < sub > q </ sub >( G ), where G is the Lie algebra whose Cartan matrix is A, is defined as the unital associative algebra with generators ( where λ is an element of the weight lattice, i. e. for all i ), and and ( for simple roots, ), subject to the following relations:

Namely, Γ ( I ) is a subgroup of the group of norm 1 elements in the quaternion algebra generated as an associative algebra by the generators i, j and relations.

In the mathematical field of representation theory, a trivial representation is a representation of a group G on which all elements of G act as the identity mapping of V. A trivial representation of an associative or Lie algebra is a ( Lie ) algebra representation for which all elements of the algebra act as the zero linear map ( endomorphism ) which sends every element of V to the zero vector.

Most associative algebras considered in abstract algebra, for instance group algebras, polynomial algebras and matrix algebras, are unital, if rings are taken to be so.

Sets and their binary operations may be classified according to their properties: for instance, if an operation is associative on a set that contains an identity element and inverses for each member of the set, the set and operation is considered to be a group.

But S < sup > 7 </ sup > is not a group in this way because octonion multiplication is not associative, nor can it be given any other continuous multiplication for which it is a group.

The Cayley – Bacharach theorem is also used to prove that the group operation on cubic elliptic curves is associative.

Thus the group operation is associative.

It is similar to a group in many ways but need not be associative.

* Any group is an associative loop and therefore a Moufang loop.

Noting that any identity matrix is a rotation matrix, and that matrix multiplication is associative, we may summarize all these properties by saying that the n × n rotation matrices form a group, which for n > 2 is non-abelian.