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Идеал (теория на пръстените) Идеал в теория на пръстените е подмножество на пръстен притежаващо специални свойства. Идеалът е конструкция подобна на нормалната група от теория на групите. Използувайки идеали може да се образуват факторпръстени, както чрез използуване на нормални групи се образуват факторгрупи. Съдържание История на понятието[редактиране | редактиране на кода] Идеалите са въведени от Рихард Дедекинд през 1876 в книгата му Лекции по теория на числата (Vorlesungen über Zahlentheorie). Появяват се като обобщение на понятието идеално число въведено от Ернст Кумер. Основни заслуги за развитието на теория на идеалите имат Давид Хилберт и Еми Ньотер. Формални определения[редактиране | редактиране на кода] Нека е пръстен. Подмножеството на , ще наричаме ляв( съответно десен) идеал, ако е непразно и са удоволетворени условията: , ще наричаме двустранен идеал или само идеал, ако е едновременно ляв и десен идеал (означава се с ). Ако пръстенът е комутативен, то всеки ляв/десен идеал е двустранен идеал. Сума на идеали дефинираме, като множеството . Собствен идеал е идеал, който не съвпада, като множество, с (бележи се с ). Примери[редактиране | редактиране на кода] - Всяка адитивна подгрупа на целите числа е идеал в пръстена на целите числа. Четните числа образуват идеал в пръстена . - В пръстена на квадратните матрици, подмножеството от матрици, чиито последен стълб е съставен само от нули, е ляв (,но не десен) идеал. Обратно, подмножеството от матрици, чиито последен ред е съставен само от нули, е десен (, но не ляв) идеал. - и са идеали във всеки пръстен . - Компактните оператори образуват идеал в пръстена на ограничените оператори. Видове идеали[редактиране | редактиране на кода] - Максимален идеал. наричаме максимален идеал ако не съществува друг собствен идеал такъв, че да съдържа като подмножество. - Прост идеал е собствен идеал, за който е изпълнено: , ако , то следва, че поне едно от a и b принадлежи на . - Примитивен идеал. Ако за всеки , ако от , следва че поне едно от a и bn принадлежи на за някое n - естествено. Всеки прост идеал е примитивен, но обратното не е вярно. - Главен идеал е идеал породен от един елемент a на комутативен пръстен с единица , бележи се . Лесно се вижда, че . - Неприводим идеал е такъв, който не може да се представи като сечение на идеали, които го съдържат като собствено подмножество. - Два идеала и се наричат комаксимални, ако за някое и . Свойства на идеалите[редактиране | редактиране на кода] - Един идеал е собствен тогава и само тогава, когато не съдържа 1. - Всеки собствен идеал се съдържа в максимален идеал. - Комутативен пръстен с 1 е поле тогава и само тогава, когато няма нетривиални идеали (различни от {0} и самият пръстен). - Всеки идеал е непразен, защото 0 принадлежи на идеала, следователно идеалът е подгрупа на адитивната група на пръстена, както и подпръстен на пръстена. - Всеки ляв/десен идеал на е ляв/десен R-подмодул на . - Сечение на леви/десни/двустранни идеали е ляв/десен/двустранен идеал. - Всеки идеал е псевдопръстен. - Идеалите на един пръстен образуват полупръстен с действия събиране и умножение на идеали.
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Prime ideal In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers.[1][2] The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together with the zero ideal. Primitive ideals are prime, and prime ideals are both primary and semiprime. Contents Prime ideals for commutative rings[edit] An ideal P of a commutative ring R is prime if it has the following two properties: - If a and b are two elements of R such that their product ab is an element of P, then a is in P or b is in P, - P is not equal to the whole ring R. This generalizes the following property of prime numbers: if p is a prime number and if p divides a product ab of two integers, then p divides a or p divides b. We can therefore say - A positive integer n is a prime number if and only if the ideal nZ is a prime ideal in Z. Examples[edit] - If R denotes the ring C[X, Y] of polynomials in two variables with complex coefficients, then the ideal generated by the polynomial Y 2 − X 3 − X − 1 is a prime ideal (see elliptic curve). - In the ring Z[X] of all polynomials with integer coefficients, the ideal generated by 2 and X is a prime ideal. It consists of all those polynomials whose constant coefficient is even. - In any ring R, a maximal ideal is an ideal M that is maximal in the set of all proper ideals of R, i.e. M is contained in exactly two ideals of R, namely M itself and the entire ring R. Every maximal ideal is in fact prime. In a principal ideal domain every nonzero prime ideal is maximal, but this is not true in general. - If M is a smooth manifold, R is the ring of smooth real functions on M, and x is a point in M, then the set of all smooth functions f with f (x) = 0 forms a prime ideal (even a maximal ideal) in R. Properties[edit] - An ideal I in the ring R (with unity) is prime if and only if the factor ring R/I is an integral domain. In particular, a commutative ring is an integral domain if and only if {0} is a prime ideal. - An ideal I is prime if and only if its set-theoretic complement is multiplicatively closed.[3] - Every nonzero ring contains at least one prime ideal (in fact it contains at least one maximal ideal), which is a direct consequence of Krull's theorem. - The set of all prime ideals (the spectrum of a ring) contains minimal elements (called minimal prime). Geometrically, these correspond to irreducible components of the spectrum. - The preimage of a prime ideal under a ring homomorphism is a prime ideal. - The sum of two prime ideals is not necessarily prime. For an example, consider the ring C[x, y] with prime ideals P = (x2 + y2 − 1) and Q = (x) (the ideals generated by x2 + y2 − 1 and x respectively). Their sum P + Q = (x2 + y2 − 1, x) = (y2 − 1, x) however is not prime: y2 − 1 = (y − 1)(y + 1) ∈ P + Q but its two factors are not. Alternatively, note that the quotient ring has zero divisors so it is not an integral domain and thus P + Q cannot be prime. - In a commutative ring R with at least two elements, if every proper ideal is prime, then the ring is a field. (If the ideal (0) is prime, then the ring R is an integral domain. If q is any non-zero element of R and the ideal (q2) is prime, then it contains q and then q is invertible.) - A nonzero principal ideal is prime if and only if it is generated by a prime element. In a UFD, every nonzero prime ideal contains a prime element. Uses[edit] One use of prime ideals occurs in algebraic geometry, where varieties are defined as the zero sets of ideals in polynomial rings. It turns out that the irreducible varieties correspond to prime ideals. In the modern abstract approach, one starts with an arbitrary commutative ring and turns the set of its prime ideals, also called its spectrum, into a topological space and can thus define generalizations of varieties called schemes, which find applications not only in geometry, but also in number theory. The introduction of prime ideals in algebraic number theory was a major step forward: it was realized that the important property of unique factorisation expressed in the fundamental theorem of arithmetic does not hold in every ring of algebraic integers, but a substitute was found when Richard Dedekind replaced elements by ideals and prime elements by prime ideals; see Dedekind domain. Prime ideals for noncommutative rings[edit] The notion of a prime ideal can be generalized to noncommutative rings by using the commutative definition "ideal-wise". Wolfgang Krull advanced this idea in 1928.[4] The following content can be found in texts such as (Goodearl 2004) and (Lam 2001). If R is a (possibly noncommutative) ring and P is an ideal in R other than R itself, we say that P is prime if for any two ideals A and B of R: - If the product of ideals AB is contained in P, then at least one of A and B is contained in P. It can be shown that this definition is equivalent to the commutative one in commutative rings. It is readily verified that if an ideal of a noncommutative ring R satisfies the commutative definition of prime, then it also satisfies the noncommutative version. An ideal P satisfying the commutative definition of prime is sometimes called a completely prime ideal to distinguish it from other merely prime ideals in the ring. Completely prime ideals are prime ideals, but the converse is not true. For example, the zero ideal in the ring of n × n matrices over a field is a prime ideal, but it is not completely prime. This is close to the historical point of view of ideals as ideal numbers, as for the ring Z "A is contained in P" is another way of saying "P divides A", and the unit ideal R represents unity. Equivalent formulations of the ideal P ≠ R being prime include the following properties: - For all a and b in R, (a)(b) ⊆ P implies a ∈ P or b ∈ P. - For any two right ideals of R, AB ⊆ P implies A ⊆ P or B ⊆ P. - For any two left ideals of R, AB ⊆ P implies A ⊆ P or B ⊆ P. - For any elements a and b of R, if aRb ⊆ P, then a ∈ P or b ∈ P. Prime ideals in commutative rings are characterized by having multiplicatively closed complements in R, and with slight modification, a similar characterization can be formulated for prime ideals in noncommutative rings. A nonempty subset S ⊆ R is called an m-system if for any a and b in S, there exists r in R such that arb is in S.[5] The following item can then be added to the list of equivalent conditions above: - The complement R\P is an m-system. Examples[edit] - Any primitive ideal is prime. - As with commutative rings, maximal ideals are prime, and also prime ideals contain minimal prime ideals. - A ring is a prime ring if and only if the zero ideal is a prime ideal, and moreover a ring is a domain if and only if the zero ideal is a completely prime ideal. - Another fact from commutative theory echoed in noncommutative theory is that if A is a nonzero R module, and P is a maximal element in the poset of annihilator ideals of submodules of A, then P is prime. Important facts[edit] - Prime avoidance lemma. If R is a commutative ring, and A is a subring (possibly without unity), and I1, ..., In is a collection of ideals of R with at most two members not prime, then if A is not contained in any Ij, it is also not contained in the union of I1, ..., In.[6] In particular, A could be an ideal of R. - If S is any m-system in R, then a lemma essentially due to Krull shows that there exists an ideal of R maximal with respect to being disjoint from S, and moreover the ideal must be prime.[7] In the case {S} = {1}, we have Krull's theorem, and this recovers the maximal ideals of R. Another prototypical m-system is the set, {x, x2, x3, x4, ...}, of all positive powers of a non-nilpotent element. - For a prime ideal P, the complement R\P has another property beyond being an m-system. If xy is in R\P, then both x and y must be in R\P, since P is an ideal. A set that contains the divisors of its elements is called saturated. - For a commutative ring R, there is a kind of converse for the previous statement: If S is any nonempty saturated and multiplicatively closed subset of R, the complement R\S is a union of prime ideals of R.[8] - The intersection of members of a descending chain of prime ideals is a prime ideal, and in a commutative ring the union of members of an ascending chain of prime ideals is a prime ideal. With Zorn's Lemma, these observations imply that the poset of prime ideals of a commutative ring (partially ordered by inclusion) has maximal and minimal elements. Connection to maximality[edit] Prime ideals can frequently be produced as maximal elements of certain collections of ideals. For example: - An ideal maximal with respect to having empty intersection with a fixed m-system is prime. - An ideal maximal among annihilators of submodules of a fixed R module M is prime. - In a commutative ring, an ideal maximal with respect to being non-principal is prime.[9] - In a commutative ring, an ideal maximal with respect to being not countably generated is prime.[10] References[edit] - ^ Dummit, David S.; Foote, Richard M. (2004). Abstract Algebra (3rd ed.). John Wiley & Sons. - ^ Lang, Serge (2002). Algebra. Graduate Texts in Mathematics. Springer. - ^ Reid, Miles (1996). Undergraduate Commutative Algebra. Cambridge University Press. - ^ Krull, Wolfgang, Primidealketten in allgemeinen Ringbereichen, Sitzungsberichte Heidelberg. Akad. Wissenschaft (1928), 7. - ^ Obviously, multiplicatively closed sets are m-systems. - ^ Jacobson Basic Algebra II, p. 390 - ^ Lam First Course in Noncommutative Rings, p. 156 - ^ Kaplansky Commutative rings, p. 2 - ^ Kaplansky Commutative rings, p. 10, Ex 10. - ^ Kaplansky Commutative rings, p. 10, Ex 11. Further reading[edit] - Goodearl, K. R.; Warfield, R. B., Jr. (2004), An introduction to noncommutative Noetherian rings, London Mathematical Society Student Texts 61 (2 ed.), Cambridge: Cambridge University Press, pp. xxiv+344, ISBN 0-521-54537-4, MR 2080008 - Jacobson, Nathan (1989), Basic algebra. II (2 ed.), New York: W. H. Freeman and Company, pp. xviii+686, ISBN 0-7167-1933-9, MR 1009787 - Kaplansky, Irving (1970), Commutative rings, Boston, Mass.: Allyn and Bacon Inc., pp. x+180, MR 0254021 - Lam, T. Y. (2001), A first course in noncommutative rings, Graduate Texts in Mathematics 131 (2nd ed.), New York: Springer-Verlag, pp. xx+385, ISBN 0-387-95183-0, MR 1838439, Zbl 0980.16001 - Lam, T. Y.; Reyes, Manuel L. (2008), A prime ideal principle in commutative algebra, J. Algebra 319 (7): 3006–3027, doi:10.1016/j.jalgebra.2007.07.016, ISSN 0021-8693, MR 2397420, Zbl 1168.13002 - Hazewinkel, Michiel, ed. (2001), "Prime ideal", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4