Unit ring
Encyclopedia
In mathematics
, a unit ring or ring with a unit is a unital ring, i.e. a ring
R with a multiplicative unit element, denoted by 1R or simply 1 if there is no risk of confusion.
s Z and all fields (Q, R, C, finite field
s Fq,...) are unit rings, and the set of all functions from a set I into a unit ring is once again a unit ring for pointwise multiplication.
Polynomial
s (with coefficients in a unit ring) and Schwartz distributions with compact support are unit rings for the convolution
product.
Most spaces of (test) functions used in analysis are rings without a unit (for pointwise multiplication), because these functions usually must decrease to 0 at infinity, so there cannot be a multiplicative unit (which must be equal to 1 everywhere).
, in a given ring R any element with a multiplicative inverse is called a unit
of the ring, i.e., the term may refer to any invertible element, not only the unit element 1R. The term ring with a unit is nevertheless well-defined, because in order to define the notion of invertible, the ring must have a unit element 1R. Thus, a ring with "any" unit is always a unital ring.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a unit ring or ring with a unit is a unital ring, i.e. a ring
Ring (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...
R with a multiplicative unit element, denoted by 1R or simply 1 if there is no risk of confusion.
Alternative definitions of a ring
Some authors (such as Lang) require a ring to have a unit by definition. In those cases, a ring without unit is called a pseudo-ring or Rng.Examples
The integerInteger
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
s Z and all fields (Q, R, C, finite field
Finite field
In abstract algebra, a finite field or Galois field is a field that contains a finite number of elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theory...
s Fq,...) are unit rings, and the set of all functions from a set I into a unit ring is once again a unit ring for pointwise multiplication.
Polynomial
Polynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...
s (with coefficients in a unit ring) and Schwartz distributions with compact support are unit rings for the convolution
Convolution
In mathematics and, in particular, functional analysis, convolution is a mathematical operation on two functions f and g, producing a third function that is typically viewed as a modified version of one of the original functions. Convolution is similar to cross-correlation...
product.
Most spaces of (test) functions used in analysis are rings without a unit (for pointwise multiplication), because these functions usually must decrease to 0 at infinity, so there cannot be a multiplicative unit (which must be equal to 1 everywhere).
"Unit" versus "Ring with unit"
In ring theoryRing theory
In abstract algebra, ring theory is the study of rings—algebraic structures in which addition and multiplication are defined and have similar properties to those familiar from the integers...
, in a given ring R any element with a multiplicative inverse is called a unit
Unit (ring theory)
In mathematics, an invertible element or a unit in a ring R refers to any element u that has an inverse element in the multiplicative monoid of R, i.e. such element v that...
of the ring, i.e., the term may refer to any invertible element, not only the unit element 1R. The term ring with a unit is nevertheless well-defined, because in order to define the notion of invertible, the ring must have a unit element 1R. Thus, a ring with "any" unit is always a unital ring.