词条 | Quotient ring |
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In ring theory, a branch of abstract algebra, a quotient ring, also known as factor ring, difference ring[1] or residue class ring, is a construction quite similar to the quotient groups of group theory and the quotient spaces of linear algebra.[2][3] One starts with a ring R and a two-sided ideal I in R, and constructs a new ring, the quotient ring {{nowrap|R / I}}, whose elements are the cosets of I in R subject to special + and ⋅ operations. Quotient rings are distinct from the so-called 'quotient field', or field of fractions, of an integral domain as well as from the more general 'rings of quotients' obtained by localization. Formal quotient ring constructionGiven a ring R and a two-sided ideal I in R, we may define an equivalence relation ~ on R as follows: a ~ b if and only if {{nowrap|a − b}} is in I. Using the ideal properties, it is not difficult to check that ~ is a congruence relation. In case a ~ b, we say that a and b are congruent modulo I. The equivalence class of the element a in R is given by [a] = a + I := { a + r : r in I }. This equivalence class is also sometimes written as a mod I and called the "residue class of a modulo I". The set of all such equivalence classes is denoted by {{nowrap|R / I}}; it becomes a ring, the factor ring or quotient ring of R modulo I, if one defines
(Here one has to check that these definitions are well-defined. Compare coset and quotient group.) The zero-element of {{nowrap|R / I}} is {{nowrap|1=(0 + I) = I}}, and the multiplicative identity is {{nowrap|(1 + I)}}. The map p from R to {{nowrap|R / I}} defined by {{nowrap|1=p(a) = a + I}} is a surjective ring homomorphism, sometimes called the natural quotient map or the canonical homomorphism. Examples
Alternative complex planesThe quotients {{nowrap|R[X] / (X)}}, {{nowrap|R[X] / (X + 1)}}, and {{nowrap|R[X] / (X − 1)}} are all isomorphic to R and gain little interest at first. But note that {{nowrap|R[X] / (X{{i sup|2}})}} is called the dual number plane in geometric algebra. It consists only of linear binomials as "remainders" after reducing an element of R[X] by X{{i sup|2}}. This alternative complex plane arises as a subalgebra whenever the algebra contains a real line and a nilpotent. Furthermore, the ring quotient {{nowrap|R[X] / (X{{i sup|2}} − 1)}} does split into {{nowrap|R[X] / (X + 1)}} and {{nowrap|R[X] / (X − 1)}}, so this ring is often viewed as the direct sum {{nowrap|R ⊕ R}}. Nevertheless, an alternative complex number {{nowrap|1=z = x + y j}} is suggested by j as a root of {{nowrap|X{{i sup|2}} − 1}}, compared to i as root of {{nowrap|1=X{{i sup|2}} + 1 = 0}}. This plane of split-complex numbers normalizes the direct sum {{nowrap|R ⊕ R}} by providing a basis {{nowrap|{1, j}{{null}}}} for 2-space where the identity of the algebra is at unit distance from the zero. With this basis a unit hyperbola may be compared to the unit circle of the ordinary complex plane. Quaternions and alternativesSuppose X and Y are two, non-commuting, indeterminates and form the free algebra {{nowrap|R{{angbr|X, Y}}}}. Then Hamilton’s quaternions of 1843 can be cast as If {{nowrap|Y{{i sup|2}} − 1}} is substituted for {{nowrap|Y{{i sup|2}} + 1}}, then one obtains the ring of split-quaternions. Substituting minus for plus in both the quadratic binomials also results in split-quaternions. The anti-commutative property {{nowrap|1=YX = −XY}} implies that XY has as its square (XY)(XY) = X(YX)Y = −X(XY)Y = −XXYY = −1. The three types of biquaternions can also be written as quotients by use of the free algebra with three indeterminates R{{angbr|X, Y, Z}} and constructing appropriate ideals. PropertiesClearly, if R is a commutative ring, then so is {{nowrap|R / I}}; the converse however is not true in general. The natural quotient map p has I as its kernel; since the kernel of every ring homomorphism is a two-sided ideal, we can state that two-sided ideals are precisely the kernels of ring homomorphisms. The intimate relationship between ring homomorphisms, kernels and quotient rings can be summarized as follows: the ring homomorphisms defined on {{nowrap|R / I}} are essentially the same as the ring homomorphisms defined on R that vanish (i.e. are zero) on I. More precisely: given a two-sided ideal I in R and a ring homomorphism {{nowrap|f : R → S}} whose kernel contains I, then there exists precisely one ring homomorphism {{nowrap|g : R / I → S}} with {{nowrap|1=gp = f}} (where p is the natural quotient map). The map g here is given by the well-defined rule {{nowrap|1=g([a]) = f(a)}} for all a in R. Indeed, this universal property can be used to define quotient rings and their natural quotient maps. As a consequence of the above, one obtains the fundamental statement: every ring homomorphism {{nowrap|f : R → S}} induces a ring isomorphism between the quotient ring {{nowrap|R / ker(f)}} and the image im(f). (See also: fundamental theorem on homomorphisms.) The ideals of R and {{nowrap|1=R / I}} are closely related: the natural quotient map provides a bijection between the two-sided ideals of R that contain I and the two-sided ideals of {{nowrap|R / I}} (the same is true for left and for right ideals). This relationship between two-sided ideal extends to a relationship between the corresponding quotient rings: if M is a two-sided ideal in R that contains I, and we write {{nowrap|M / I}} for the corresponding ideal in {{nowrap|R / I}} (i.e. {{nowrap|1=M / I = p(M)}}), the quotient rings {{nowrap|1=R / M}} and {{nowrap|(R / I) / (M / I)}} are naturally isomorphic via the (well-defined!) mapping {{nowrap|a + M ↦ (a + I) + M / I}}. In commutative algebra and algebraic geometry, the following statement is often used: If {{nowrap|1=R ≠ {0}{{null}}}} is a commutative ring and I is a maximal ideal, then the quotient ring {{nowrap|R / I}} is a field; if I is only a prime ideal, then {{nowrap|R / I}} is only an integral domain. A number of similar statements relate properties of the ideal I to properties of the quotient ring {{nowrap|R / I}}. The Chinese remainder theorem states that, if the ideal I is the intersection (or equivalently, the product) of pairwise coprime ideals I1, ..., Ik, then the quotient ring {{nowrap|R / I}} is isomorphic to the product of the quotient rings {{nowrap|R / Ip}}, {{nowrap|1=p = 1, ..., k}}. See also
Notes1. ^{{cite book | authorlink=Nathan Jacobson | last1=Jacobson | first1=Nathan | title=Structure of Rings | publisher=American Mathematical Soc. | year=1984 | edition=revised | isbn=0-821-87470-5}} 2. ^{{cite book | last1=Dummit | first1=David S. | last2=Foote | first2=Richard M. | title=Abstract Algebra | publisher=John Wiley & Sons | year=2004 | edition=3rd | isbn=0-471-43334-9}} 3. ^{{cite book | last=Lang | first=Serge | authorlink=Serge Lang | title=Algebra | publisher=Springer | series=Graduate Texts in Mathematics | year=2002 | isbn=0-387-95385-X}} Further references
External links
2 : Quotient objects|Ring theory |
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