Unit ring

In mathematics, a unit ring or ring with a unit is a unital ring, i.e. a ring R with a multiplicative identity element, denoted by 1_R 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 integers Z and all fields (Q, R, C, finite fields F_q,...) 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.

Polynomials (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).

"Unit" versus "Ring with unit"

In ring theory, 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 1_R. 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 1_R. Thus, a ring with "any" unit is always a unital ring.

See also

• Glossary of ring theory
• Semiring
• Unital algebra

References

• Wilder, Raymond L. (1965), Introduction to the Foundations of Mathematics, John Wiley and Sons, New York, NY. Uses the terminology ring with a unit in the definition of rings on page 176.
• Weisstein, Eric W. "Unit Ring". MathWorld.