In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy four conditions called the group axioms, namely closure, associativity, identity and invertibility. Many familiar mathematical structures such as number systems obey these axioms: for example, the integers endowed with the addition operation form a group.
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Matrix (mathematics)
In mathematics, a matrix (plural matrices, or less commonly matrixes) is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements is Matrices of the same size can be added or subtracted element by element. The rule for matrix multiplication is more complicated, and two matrices can be multiplied only when the number of columns in the first equals the number of rows in the second.
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Division ring
In abstract algebra, a division ring, also called a skew field, is a ring in which division is possible. Specifically, it is a non-trivial ring in which every non-zero element a has a multiplicative inverse, i.e. , an element x with {{{1}}}. Stated differently, a ring is a division ring if and only if the group of units is the set of all non-zero elements. Division rings differ from fields only in that their multiplication is not required to be commutative.
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Ring (mathematics)
In mathematics, a ring is an algebraic structure which generalizes the main properties of the addition and the multiplication of integers, real and complex numbers, as well as that of square matrices.
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Element (mathematics)
In mathematics, an element or member of a set is any one of the distinct objects that make up that set.
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Field (mathematics)
In abstract algebra, a field is a ring whose nonzero elements form a commutative group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms. The most commonly used fields are the field of real numbers, the field of complex numbers, and the field of rational numbers, but there are also finite fields, fields of functions, various algebraic number fields, p-adic fields, and so forth.
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