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Centrally Essential Ring

A centrally essential ring is a ring R with a nonzero multiplicative identity such that, for every nonzero a in R, there are nonzero elements x,y in the ring center with ax=y. Thus multiplying a by a suitable nonzero central element produces a nonzero central element (Markov and Tuganbaev 2018). Every commutative ring with a nonzero multiplicative identity is centrally essential, but the converse fails.

Abyzov et al. (2026) proved that the smallest finite ring that is centrally essential and noncommutative has 64 elements. They also exhibited nonisomorphic examples of this smallest order.

See also

Commutative Ring, Ring Center

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References

Abyzov, A.; Lyubimtsev, O.; and Tuganbaev, A. "Finite Non-Commutative Centrally Essential Rings of Minimal Order." 26 Oct 2026. https://arxiv.org/abs/2610.36174.Markov, V. T. and Tuganbaev, A. A. "Centrally Essential Group Algebras." J. Algebra 512, 109-118, 2018. https://doi.org/10.1016/j.jalgebra.2018.07.009.

Cite this as:

Weisstein, Eric W. "Centrally Essential Ring." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/CentrallyEssentialRing.html

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