Ring Theory

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10 domain-specific abstractions whose origin domain is Ring Theory.

• Domain (ring theory) — A nonzero ring with no nonzero left or right zero divisors.
• Euclidean domain — An integral domain equipped with a Euclidean function that supports division with remainder of strictly smaller value and therefore the Euclidean algorithm.
• Gelfand ring — A ring satisfying separation conditions on distinct maximal ideals that generalize topological features of Gelfand duality.
• Idempotent (ring theory) — A ring element e satisfying e²=e, whose multiplication acts as a projection and whose presence can encode decompositions of rings, modules and spectra.
• Invariant basis number — A ring property ensuring that isomorphic finitely generated free modules have the same finite rank, so basis cardinality is well defined.
• Köthe conjecture — The open ring-theoretic conjecture that the sum of two nil left ideals is nil, equivalently that a ring with no nonzero nil ideal has no nonzero nil one-sided ideal.
• Perfect ring — A ring for which every module on the specified side has a projective cover, with equivalent chain and radical conditions under Bass's theorem.
• Polynomial identity ring — A ring on which some nonzero noncommutative polynomial vanishes under every substitution of ring elements.
• Primitive ring — A ring admitting a faithful simple left module or, separately, a faithful simple right module.
• Radical of a ring — An ideal-valued construction that isolates elements regarded as structurally degenerate under a chosen radical theory and yields a semisimple quotient.