nLab reduced local ring

 Definition

A reduced local ring is a commutative ring which is both a local ring and a reduced ring, hence a commutative ring such that:

• The ring is nontrivial $0 \neq 1$;

• if the sum of two elements $x + y$ is equal to $1$, then $x$ is invertible or $y$ is invertible;

• every nilpotent element is equal to zero; equivalently, every element which squares to zero is equal to zero.

 Examples

• Every Heyting field is a reduced local ring which is also a Artinian ring.

• Given a discrete field $K$, the ring of power series $K[[x]]$ is a reduced local ring which is not a Heyting field.

Properties

The theory of reduced local rings is a coherent theory.

 See also

• ordered reduced local ring

References

For a division algorithm for polynomials over reduced local rings:

• Fred Richman, A division algorithm. Journal of Algebra and Its Applications, Vol. 04, No. 04, pp. 441-449 (2005) [doi:10.1142/S0219498805001289]

commutative ring	reduced ring	integral domain
local ring	reduced local ring	local integral domain
Artinian ring	semisimple ring	field
Weil ring	field	field