Opposite ring
In algebra, the opposite of a ring is another ring with the same elements and addition operation, but with the multiplication performed in the reverse order.
More precisely, the opposite of a ring (R, +, ·) is the ring (R, +, ∗) whose multiplication ∗ is defined by a ∗ b = b · a. Ring addition is per definition commutative.
Properties
Two rings R1 and R2 are isomorphic if and only if their corresponding opposite rings are isomorphic. The opposite of the opposite of a ring is isomorphic to that ring. A ring and its opposite ring are anti-isomorphic.
A commutative ring is always equal to its opposite ring. A non-commutative ring may or may not be isomorphic to its opposite ring.
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*-algebra
In mathematics, and more specifically in abstract algebra, a *-algebra (or involutive algebra) is a mathematical structure consisting of two involutive rings R and A, where R is commutative and A has the structure of an associative algebra over R. Involutive algebras generalize the idea of a number system equipped with conjugation, for example the complex numbers and complex conjugation, matrices over the complex numbers and conjugate transpose, and linear operators over a Hilbert space and Hermitian adjoints.
Terminology
*-ring
In mathematics, a *-ring is a ring with a map * : A → A that is an antiautomorphism and an involution.
More precisely, * is required to satisfy the following properties:
for all x, y in A.
This is also called an involutive ring, involutory ring, and ring with involution. Note that the third axiom is actually redundant, because the second and fourth axioms imply 1* is also a multiplicative identity, and identities are unique.
Algebra
Algebra (from Arabic "al-jabr" meaning "reunion of broken parts") is one of the broad parts of mathematics, together with number theory, geometry and analysis. In its most general form, algebra is the study of mathematical symbols and the rules for manipulating these symbols; it is a unifying thread of almost all of mathematics. As such, it includes everything from elementary equation solving to the study of abstractions such as groups, rings, and fields. The more basic parts of algebra are called elementary algebra, the more abstract parts are called abstract algebra or modern algebra. Elementary algebra is generally considered to be essential for any study of mathematics, science, or engineering, as well as such applications as medicine and economics. Abstract algebra is a major area in advanced mathematics, studied primarily by professional mathematicians. Much early work in algebra, as the Arabic origin of its name suggests, was done in the Middle East, by mathematicians such as al-Khwārizmī (780 – 850) and Omar Khayyam (1048–1131).
†-algebra
A †-algebra (or, more explicitly, a †-closed algebra) is the name occasionally used in physics for a finite-dimensional C*-algebra. The dagger, †, is used in the name because physicists typically use the symbol to denote a hermitian adjoint, and are often not worried about the subtleties associated with an infinite number of dimensions. (Mathematicians usually use the asterisk, *, to denote the hermitian adjoint.) †-algebras feature prominently in quantum mechanics, and especially quantum information science.
References
Opposite
Opposite may refer to:
Basic meanings
Film and TV
Music
Albums
Songs
See also
Additive inverse
In mathematics, the additive inverse of a number a is the number that, when added to a, yields zero. This number is also known as the opposite (number),sign change, and negation. For a real number, it reverses its sign: the opposite to a positive number is negative, and the opposite to a negative number is positive. Zero is the additive inverse of itself.
The additive inverse of a is denoted by unary minus: −a (see the discussion below). For example, the additive inverse of 7 is −7, because 7 + (−7) = 0, and the additive inverse of −0.3 is 0.3, because −0.3 + 0.3 = 0 .
The additive inverse is defined as its inverse element under the binary operation of addition (see the discussion below), which allows a broad generalization to mathematical objects other than numbers. As for any inverse operation, double additive inverse has no effect: −(−x) = x.
Common examples
For a number and, generally, in any ring, the additive inverse can be calculated using multiplication by −1; that is, −n = −1 × n . Examples of rings of numbers are integers, rational numbers, real numbers, and complex number.
Opposite (song)
"Opposite" is a song by Scottish alternative rock band Biffy Clyro, released as the third single from the band's sixth studio album, Opposites (2013), on June 24, 2013.
It made number 49 on the Official UK singles chart.
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External links
Podcasts:
