Definition:Null Ring

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A ring with one element is called the null ring.

That is, the null ring is $\struct {\set {0_R}, +, \circ}$, where ring addition and the ring product are defined as:

\(\ds 0_R + 0_R\) \(=\) \(\ds 0_R\)
\(\ds 0_R \circ 0_R\) \(=\) \(\ds 0_R\)

Also known as

Some authors refer to this as the zero ring, others as the degenerate ring.

Still others refer to it as the trivial ring, but this term has been defined differently elsewhere.

Also see

  • Results about the null ring can be found here.