Examples
The integers Z and all fields (Q, R, C, finite fields 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.
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).
Read more about this topic: Unit Ring
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