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List Of Abstract Algebra Topics
Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. The phrase abstract algebra was coined at the turn of the 20th century to distinguish this area from what was normally referred to as algebra, the study of the rules for manipulating formulae and algebraic expressions involving unknowns and real or complex numbers, often now called ''elementary algebra''. The distinction is rarely made in more recent writings. Basic language Algebraic structures are defined primarily as sets with ''operations''. *Algebraic structure **Subobjects: subgroup, subring, subalgebra, submodule etc. *Binary operation ** Closure of an operation **Associative property **Distributive property **Commutative property *Unary operator **Additive inverse, multiplicative inverse, inverse element ***Identity element ***Cancellation property * Finitary operation **Arity Structure preserving maps called ''homomo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abstract Algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term ''abstract algebra'' was coined in the early 20th century to distinguish this area of study from older parts of algebra, and more specifically from elementary algebra, the use of variables to represent numbers in computation and reasoning. Algebraic structures, with their associated homomorphisms, form mathematical categories. Category theory is a formalism that allows a unified way for expressing properties and constructions that are similar for various structures. Universal algebra is a related subject that studies types of algebraic structures as single objects. For example, the structure of groups is a single object in universal algebra, which is called the '' variety of groups''. History Before the nineteenth century, alge ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Subalgebra
In mathematics, a subalgebra is a subset of an algebra, closed under all its operations, and carrying the induced operations. "Algebra", when referring to a structure, often means a vector space or module equipped with an additional bilinear operation. Algebras in universal algebra are far more general: they are a common generalisation of ''all'' algebraic structures. "Subalgebra" can refer to either case. Subalgebras for algebras over a ring or field A subalgebra of an algebra over a commutative ring or field is a vector subspace which is closed under the multiplication of vectors. The restriction of the algebra multiplication makes it an algebra over the same ring or field. This notion also applies to most specializations, where the multiplication must satisfy additional properties, e.g. to associative algebras or to Lie algebras. Only for unital algebras is there a stronger notion, of unital subalgebra, for which it is also required that the unit of the subalgebra be the u ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finitary Operation
In mathematics, an operation is a function which takes zero or more input values (also called "''operands''" or "arguments") to a well-defined output value. The number of operands is the arity of the operation. The most commonly studied operations are binary operations (i.e., operations of arity 2), such as addition and multiplication, and unary operations (i.e., operations of arity 1), such as additive inverse and multiplicative inverse. An operation of arity zero, or nullary operation, is a constant. The mixed product is an example of an operation of arity 3, also called ternary operation. Generally, the arity is taken to be finite. However, infinitary operations are sometimes considered, in which case the "usual" operations of finite arity are called finitary operations. A partial operation is defined similarly to an operation, but with a partial function in place of a function. Types of operation There are two common types of operations: unary and b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cancellation Property
In mathematics, the notion of cancellative is a generalization of the notion of invertible. An element ''a'' in a magma has the left cancellation property (or is left-cancellative) if for all ''b'' and ''c'' in ''M'', always implies that . An element ''a'' in a magma has the right cancellation property (or is right-cancellative) if for all ''b'' and ''c'' in ''M'', always implies that . An element ''a'' in a magma has the two-sided cancellation property (or is cancellative) if it is both left- and right-cancellative. A magma has the left cancellation property (or is left-cancellative) if all ''a'' in the magma are left cancellative, and similar definitions apply for the right cancellative or two-sided cancellative properties. A left-invertible element is left-cancellative, and analogously for right and two-sided. For example, every quasigroup, and thus every group, is cancellative. Interpretation To say that an element ''a'' in a magma is left-cancellative, is to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Identity Element
In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures such as groups and rings. The term ''identity element'' is often shortened to ''identity'' (as in the case of additive identity and multiplicative identity) when there is no possibility of confusion, but the identity implicitly depends on the binary operation it is associated with. Definitions Let be a set equipped with a binary operation ∗. Then an element of is called a if for all in , and a if for all in . If is both a left identity and a right identity, then it is called a , or simply an . An identity with respect to addition is called an (often denoted as 0) and an identity with respect to multiplication is called a (often denoted as 1). These need not be ordinary add ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inverse Element
In mathematics, the concept of an inverse element generalises the concepts of opposite () and reciprocal () of numbers. Given an operation denoted here , and an identity element denoted , if , one says that is a left inverse of , and that is a right inverse of . (An identity element is an element such that and for all and for which the left-hand sides are defined.) When the operation is associative, if an element has both a left inverse and a right inverse, then these two inverses are equal and unique; they are called the ''inverse element'' or simply the ''inverse''. Often an adjective is added for specifying the operation, such as in additive inverse, multiplicative inverse, and functional inverse. In this case (associative operation), an invertible element is an element that has an inverse. Inverses are commonly used in groupswhere every element is invertible, and ringswhere invertible elements are also called units. They are also commonly used for operatio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multiplicative Inverse
In mathematics, a multiplicative inverse or reciprocal for a number ''x'', denoted by 1/''x'' or ''x''−1, is a number which when multiplied by ''x'' yields the multiplicative identity, 1. The multiplicative inverse of a fraction ''a''/''b'' is ''b''/''a''. For the multiplicative inverse of a real number, divide 1 by the number. For example, the reciprocal of 5 is one fifth (1/5 or 0.2), and the reciprocal of 0.25 is 1 divided by 0.25, or 4. The reciprocal function, the function ''f''(''x'') that maps ''x'' to 1/''x'', is one of the simplest examples of a function which is its own inverse (an involution). Multiplying by a number is the same as dividing by its reciprocal and vice versa. For example, multiplication by 4/5 (or 0.8) will give the same result as division by 5/4 (or 1.25). Therefore, multiplication by a number followed by multiplication by its reciprocal yields the original number (since the product of the number and its reciprocal is 1). The term ''reciprocal' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Additive Inverse
In mathematics, the additive inverse of a number is the number that, when added to , yields zero. This number is also known as the opposite (number), sign change, and negation. For a real number, it reverses its sign: the additive inverse (opposite number) of a positive number is negative, and the additive inverse of a negative number is positive. Zero is the additive inverse of itself. The additive inverse of is denoted by unary minus: (see also below). For example, the additive inverse of 7 is −7, because , and the additive inverse of −0.3 is 0.3, because . Similarly, the additive inverse of is which can be simplified to . The additive inverse of is , because . The additive inverse is defined as its inverse element under the binary operation of addition (see also below), which allows a broad generalization to mathematical objects other than numbers. As for any inverse operation, double additive inverse has no net effect: . Common examples For a number (and m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unary Operator
In mathematics, an unary operation is an operation with only one operand, i.e. a single input. This is in contrast to binary operations, which use two operands. An example is any function , where is a set. The function is a unary operation on . Common notations are prefix notation (e.g. ¬, −), postfix notation (e.g. factorial ), functional notation (e.g. or ), and superscripts (e.g. transpose ). Other notations exist as well, for example, in the case of the square root, a horizontal bar extending the square root sign over the argument can indicate the extent of the argument. Examples Unary negative and positive As unary operations have only one operand they are evaluated before other operations containing them. Here is an example using negation: :3 − −2 Here, the first '−' represents the binary subtraction operation, while the second '−' represents the unary negation of the 2 (or '−2' could be taken to mean the integer −2). Therefore, the expressi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Commutative Property
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of the property that says something like or , the property can also be used in more advanced settings. The name is needed because there are operations, such as division and subtraction, that do not have it (for example, ); such operations are ''not'' commutative, and so are referred to as ''noncommutative operations''. The idea that simple operations, such as the multiplication and addition of numbers, are commutative was for many years implicitly assumed. Thus, this property was not named until the 19th century, when mathematics started to become formalized. A similar property exists for binary relations; a binary relation is said to be symmetric if the relation applies regardless of the order of its operands; for example, equality is sy ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Distributive Property
In mathematics, the distributive property of binary operations generalizes the distributive law, which asserts that the equality x \cdot (y + z) = x \cdot y + x \cdot z is always true in elementary algebra. For example, in elementary arithmetic, one has 2 \cdot (1 + 3) = (2 \cdot 1) + (2 \cdot 3). One says that multiplication ''distributes'' over addition. This basic property of numbers is part of the definition of most algebraic structures that have two operations called addition and multiplication, such as complex numbers, polynomials, matrices, rings, and fields. It is also encountered in Boolean algebra and mathematical logic, where each of the logical and (denoted \,\land\,) and the logical or (denoted \,\lor\,) distributes over the other. Definition Given a set S and two binary operators \,*\, and \,+\, on S, *the operation \,*\, is over (or with respect to) \,+\, if, given any elements x, y, \text z of S, x * (y + z) = (x * y) + (x * z); *the operation \,*\, i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Associative Property
In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement for expressions in logical proofs. 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. That is (after rewriting the expression with parentheses and in infix notation if necessary), rearranging the parentheses in such an expression will not change its value. Consider the following equations: \begin (2 + 3) + 4 &= 2 + (3 + 4) = 9 \,\\ 2 \times (3 \times 4) &= (2 \times 3) \times 4 = 24 . \end Even though the parentheses were rearranged on each line, the values of the expressions were not altered. Since this holds true when performing addition and multiplication on any r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
