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Stably Finite Ring
In mathematics, particularly in abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are set (mathematics), sets with specific operation (mathematics), operations acting on their elements. Algebraic structur ..., a ring (mathematics), ring ''R'' is said to be stably finite (or weakly finite) if, for all square matrices ''A'' and ''B'' of the same size with entries in ''R'', ''AB'' = 1 implies ''BA'' = 1. This is a stronger property for a ring than having the invariant basis number (IBN) property. Namely, any trivial ring, nontrivialA trivial ring is stably finite but doesn't have IBN. stably finite ring has IBN. Commutative rings, Noetherian rings and Artinian rings are stably finite. Subrings of stably finite rings and matrix rings over stably finite rings are stably finite. A ring satisfying Klein's nilpotence condition is stably finite. Notes References Ring ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...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, which are set (mathematics), sets with specific operation (mathematics), operations acting on their elements. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathematics), fields, module (mathematics), modules, vector spaces, lattice (order), lattices, and algebra over a field, algebras over a field. The term ''abstract algebra'' was coined in the early 20th century to distinguish it from older parts of algebra, and more specifically from elementary algebra, the use of variable (mathematics), variables to represent numbers in computation and reasoning. The abstract perspective on algebra has become so fundamental to advanced mathematics that it is simply called "algebra", while the term "abstract algebra" is seldom used except in mathematical education, pedagogy. Algebraic structures, with their associated homomorphisms, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ring (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set with two binary operations called ''addition'' and ''multiplication'', which obey the same basic laws as addition and multiplication of integers, except that multiplication in a ring does not need to be commutative. Ring elements may be numbers such as integers or complex numbers, but they may also be non-numerical objects such as polynomials, square matrices, functions, and power series. A ''ring'' may be defined as a set that is endowed with two binary operations called ''addition'' and ''multiplication'' such that the ring is an abelian group with respect to the addition operator, and the multiplication operator is associative, is distributive over the addition operation, and has a multiplicative identity element. (Some authors apply the term ''ring'' to a further generalization, often called a '' rng'', that omits the requirement for a multiplicative identity, and instead call the structure defi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Square Matrices
In mathematics, a square matrix is a matrix with the same number of rows and columns. An ''n''-by-''n'' matrix is known as a square matrix of order Any two square matrices of the same order can be added and multiplied. Square matrices are often used to represent simple linear transformations, such as shearing or rotation. For example, if R is a square matrix representing a rotation (rotation matrix) and \mathbf is a column vector describing the position of a point in space, the product R\mathbf yields another column vector describing the position of that point after that rotation. If \mathbf is a row vector, the same transformation can be obtained using where R^ is the transpose of Main diagonal The entries a_ () form the main diagonal of a square matrix. They lie on the imaginary line which runs from the top left corner to the bottom right corner of the matrix. For instance, the main diagonal of the 4×4 matrix above contains the elements , , , . The diagonal of a square m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Invariant Basis Number
In the mathematical field of ring theory, a ring ''R'' has the invariant basis number (IBN) property if all finitely generated free modules over ''R'' have a well-defined rank. In the case of fields, the IBN property is the fact that finite-dimensional vector spaces have a unique dimension. Definition A ring ''R'' has invariant basis number (IBN) if for all positive integers ''m'' and ''n'', ''R''''m'' isomorphic to ''R''''n'' (as left ''R''-modules) implies that . Equivalently, this means there do not exist distinct positive integers ''m'' and ''n'' such that ''R''''m'' is isomorphic to ''R''''n''. Rephrasing the definition of invariant basis number in terms of matrices, it says that, whenever ''A'' is an ''m''-by-''n'' matrix over ''R'' and ''B'' is an ''n''-by-''m'' matrix over ''R'' such that and , then . This form reveals that the definition is left–right symmetric, so it makes no difference whether we define IBN in terms of left or right modules; the two definitions ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trivial Ring
In ring theory, a branch of mathematics, the zero ring or trivial ring is the unique ring (up to isomorphism) consisting of one element. (Less commonly, the term "zero ring" is used to refer to any rng of square zero, i.e., a rng in which for all ''x'' and ''y''. This article refers to the one-element ring.) In the category of rings, the zero ring is the terminal object, whereas the ring of integers Z is the initial object. Definition The zero ring, denoted or simply 0, consists of the one-element set with the operations + and · defined such that 0 + 0 = 0 and 0 · 0 = 0. Properties * The zero ring is the unique ring in which the additive identity 0 and multiplicative identity 1 coincide. (Proof: If in a ring ''R'', then for all ''r'' in ''R'', we have . The proof of the last equality is found here.) * The zero ring is commutative. * The element 0 in the zero ring is a unit, serving as its own multiplicative inverse. * The unit group of the zero ring is the tr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Commutative Ring
In mathematics, a commutative ring is a Ring (mathematics), ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not specific to commutative rings. This distinction results from the high number of fundamental properties of commutative rings that do not extend to noncommutative rings. Commutative rings appear in the following chain of subclass (set theory), class inclusions: Definition and first examples Definition A ''ring'' is a Set (mathematics), set R equipped with two binary operations, i.e. operations combining any two elements of the ring to a third. They are called ''addition'' and ''multiplication'' and commonly denoted by "+" and "\cdot"; e.g. a+b and a \cdot b. To form a ring these two operations have to satisfy a number of properties: the ring has to be an abelian group under addition as well as a monoid under m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Noetherian Ring
In mathematics, a Noetherian ring is a ring that satisfies the ascending chain condition on left and right ideals. If the chain condition is satisfied only for left ideals or for right ideals, then the ring is said left-Noetherian or right-Noetherian respectively. Formally, every increasing sequence I_1\subseteq I_2 \subseteq I_3 \subseteq \cdots of left (or right) ideals has a largest element; that is, there exists an n such that I_=I_=\cdots. Equivalently, a ring is left-Noetherian (respectively right-Noetherian) if every left ideal (respectively right-ideal) is finitely generated. A ring is Noetherian if it is both left- and right-Noetherian. Noetherian rings are fundamental in both commutative and noncommutative ring theory since many rings that are encountered in mathematics are Noetherian (in particular the ring of integers, polynomial rings, and rings of algebraic integers in number fields), and many general theorems on rings rely heavily on the Noetherian property ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Artinian Ring
In mathematics, specifically abstract algebra, an Artinian ring (sometimes Artin ring) is a ring that satisfies the descending chain condition on (one-sided) ideals; that is, there is no infinite descending sequence of ideals. Artinian rings are named after Emil Artin, who first discovered that the descending chain condition for ideals simultaneously generalizes finite rings and rings that are finite-dimensional vector spaces over fields. The definition of Artinian rings may be restated by interchanging the descending chain condition with an equivalent notion: the minimum condition. Precisely, a ring is left Artinian if it satisfies the descending chain condition on left ideals, right Artinian if it satisfies the descending chain condition on right ideals, and Artinian or two-sided Artinian if it is both left and right Artinian. For commutative rings the left and right definitions coincide, but in general they are distinct from each other. The Wedderburn–Artin theorem ch ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Subring
In mathematics, a subring of a ring is a subset of that is itself a ring when binary operations of addition and multiplication on ''R'' are restricted to the subset, and that shares the same multiplicative identity as .In general, not all subsets of a ring are rings. Definition A subring of a ring is a subset of that preserves the structure of the ring, i.e. a ring with . Equivalently, it is both a subgroup of and a submonoid of . Equivalently, is a subring if and only if it contains the multiplicative identity of , and is closed under multiplication and subtraction. This is sometimes known as the ''subring test''. Variations Some mathematicians define rings without requiring the existence of a multiplicative identity (see '). In this case, a subring of is a subset of that is a ring for the operations of (this does imply it contains the additive identity of ). This alternate definition gives a strictly weaker condition, even for rings that do have a mult ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matrix Ring
In abstract algebra, a matrix ring is a set of matrices with entries in a ring ''R'' that form a ring under matrix addition and matrix multiplication. The set of all matrices with entries in ''R'' is a matrix ring denoted M''n''(''R'') (alternative notations: Mat''n''(''R'') and ). Some sets of infinite matrices form infinite matrix rings. A subring of a matrix ring is again a matrix ring. Over a rng, one can form matrix rngs. When ''R'' is a commutative ring, the matrix ring M''n''(''R'') is an associative algebra over ''R'', and may be called a matrix algebra. In this setting, if ''M'' is a matrix and ''r'' is in ''R'', then the matrix ''rM'' is the matrix ''M'' with each of its entries multiplied by ''r''. Examples * The set of all square matrices over ''R'', denoted M''n''(''R''). This is sometimes called the "full ring of ''n''-by-''n'' matrices". * The set of all upper triangular matrices over ''R''. * The set of all lower triangular matrices over ''R''. * The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
