Revision #258 → #1485 · back to history
addedPrime ideal (informal)d00faac65457
addedPrime ideals of the integers376aa89dc806
addedPrimitive ideals are prime; prime ideals are primary and semiprime55b2a15c6202
addedPrime ideal of a commutative ringfff6fa8ca2f5
addedEuclid's lemma for prime numbers3c47ee737d2a
addedn is a prime number iff (n) is a prime ideal in Z992a18d99dac
addedPrime spectrum of a commutative ring19ee7b83c4c7
addedAffine scheme (spectrum with topology and structure sheaf)23d7e338223b
addedAlternative definition of prime ideal (contrapositive form)235b056281c8
addedEquivalence of alternative definition via contrapositive398655ad7a6c
addedEven integers form a prime ideal of Zc310003b70ee
addedPrime element of an integral domain generates a principal prime ideal07966f22ef84
addedIrreducible polynomial over a field generates a prime ideal8cc2ebe92e8e
addedEisenstein's criterion for irreducibility in polynomial rings over UFDsddd56492e000
addedIdeal generated by Y^2 - X^3 - X - 1 in C[X,Y] is prime (elliptic curve)c2727a181ce8
addedIdeal (2, X) in Z[X] is a prime ideal8e3574cdcda9
addedMaximal idealebfeecdff67d
addedEvery maximal ideal is prime5655ada6f99c
addedIn a PID every nonzero prime ideal is maximal3cf9ba61222d
addedHilbert's Nullstellensatz: form of maximal ideals in k[x_1,...,x_n]384016d21778
addedVanishing functions at a point on a smooth manifold form a (maximal) prime ideal15c5fa5db677
addedNon-example: quotient ring is not an integral domain665f378521c3
addedNon-example: factorization yields zero divisors in the quotient (via CRT)d4c3cfa44144
addedNon-example: the ideal is not prime8d2ea21ccd4e
addedNon-example: product lies in ideal but neither factor does6b4401436c23
addedI is prime iff R/I is an integral domainbe389f87c4fa
addedCommutative ring is an integral domain iff (0) is prime68790a1d8dd2
addedZero ring has no prime idealsbf5a6c30f242
addedI is prime iff its complement is multiplicatively closed57b43c087226
addedEvery nonzero ring contains a prime (and maximal) ideal (Krull's theorem)e2c7c4817a1b
addedKrull's lemma: ideal maximal disjoint from a multiplicatively closed set is primefd69205cef7c
addedPositive powers of a non-nilpotent element form a prototypical m-system4c01777a15a5
addedPreimage of a prime ideal under a ring homomorphism is primeafc1012cc5dc
addedPreimage of a maximal ideal need not be maximal32fcecbe62bb
addedMinimal prime ideal45e008de18a0
addedMinimal primes correspond to irreducible components of the spectrum104877ce420f
addedSum of two prime ideals is not necessarily primee4ce497013e4
addedNot every irreducible ideal is primef4a664e3c056
addedCommutative ring in which every proper ideal is prime is a field4285df8c8e33
addedNonzero principal ideal is prime iff generated by a prime elementb93b3f8eaa28
addedIn a UFD, every nonzero prime ideal contains a prime element53fe4d27cdfc
addedIrreducible varieties correspond to prime ideals2a8fa27f399d
addedScheme (generalization of variety via spectrum of prime ideals)4e9ffd002c04
addedUnique factorization fails in some rings of algebraic integers7094368fa499
addedPrime ideal in a noncommutative ring1ed62af8ba9e
addedNoncommutative definition is equivalent to the commutative one in commutative rings847c6068fa04
addedCommutative-prime ideal in a noncommutative ring is also prime in the noncommutative sensee05405b96612
addedCompletely prime ideal32e5eb863ff7
addedCompletely prime ideals are prime, but the converse is false4374f3cea5cc
addedZero ideal of n x n matrix ring over a field is prime but not completely prime3177c248ec8b
addedEquivalent formulations of P being prime (noncommutative ring)782f2a512195
addedEquivalent formulation: principal-ideal-product condition125d08be8cd3
addedEquivalent formulation: right-ideal product condition4739ef153163
addedEquivalent formulation: left-ideal product condition6affdbffb0f0
addedEquivalent formulation: aRb subset of P condition693993d4e448
addedm-system52421c49accb
addedComplement of a prime ideal is an m-system5072c338b990
addedAny primitive ideal is prime137b8606e9af
addedMaximal ideals are prime; prime ideals contain minimal primes (noncommutative)2d3ac9f0d51f
addedRing is prime iff (0) is prime; ring is a domain iff (0) is completely primec80262522a77
addedMaximal annihilator of a submodule of a nonzero module is prime51e7f94c16e0
addedPrime avoidance lemma493f49cb771a
addedPrime not containing any I_j does not contain their intersectionbd00f4882c49
addedKrull's lemma (m-system version): ideal maximal disjoint from m-system is prime2cd23d0df6fd
addedPositive powers of a non-nilpotent element form a prototypical m-systemc1b2c6a73d19
addedSaturated subset (closed under divisors)c9a60adde675
addedComplement of a prime ideal is saturated and multiplicatively closedeae49ef0ffa0
addedComplement of a saturated multiplicatively closed set is a union of prime ideals960a6f0be859
addedIntersection (and in commutative case, union) of a chain of prime ideals is prime9b1a9351e2e2
addedPoset of prime ideals has maximal and minimal elements (via Zorn)e7b527806445
addedPrime ideals arise as maximal elements of certain collections of ideals51a57eb9e834
addedIdeal maximal w.r.t. empty intersection with a fixed m-system is primed9ccd67bf6ea
addedIdeal maximal among annihilators of submodules of a fixed R-module is prime2edd5f7a5f78
addedIn a commutative ring, an ideal maximal w.r.t. being non-principal is prime20f9e46b4f83
addedIn a commutative ring, an ideal maximal w.r.t. being not countably generated is primee2b3e9ce9517