Unit ring

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 1_"R" or simply 1 if there is no risk of confusion.

Alternative definitions of a ring

Some authors (such as Herstein) require a ring to have a unit by definition. In those cases, a ring without unit is called a "pseudoring" 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"

Notice that a unit in ring theory is 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.

ee also

*Glossary of ring theory
*Ring (mathematics)
*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.
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