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# Topic: Ring homomorphism

 PlanetMath: ring homomorphism A ring isomorphism is a ring homomorphism which is a bijection. A ring monomorphism (respectively, ring epimorphism) is a ring homomorphism which is an injection (respectively, surjection). This is version 7 of ring homomorphism, born on 2001-10-19, modified 2006-10-22. planetmath.org /encyclopedia/RingHomomorphism.html   (168 words)

 NationMaster - Encyclopedia: Ring homomorphism   (Site not responding. Last check: 2007-10-26) In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation obeys the commutative law. If R[X] denotes the ring of all polynomials in the variable X with coefficients in the real numbers R, and C denotes the complex numbers, then the function f : R[X] → C defined by f(p) = p(i) (substitute the imaginary unit i for the variable X in the polynomial p) is a surjective ring homomorphism. In abstract algebra, a monoid ring is a procedure which constructs a new ring from a given ring and a monoid. www.nationmaster.com /encyclopedia/Ring-homomorphism   (1650 words)

 PlanetMath: local ring The name comes from the fact that these rings are important in the study of the local behavior of varieties and manifolds: the ring of function germs at a point is always local. Rings of formal power series over a field are local, even in several variables. In that case, the ring also has a unique maximal right ideal, and the two ideals coincide with the ring's Jacobson radical, which in this case consists precisely of the non-units in the ring. planetmath.org /encyclopedia/LocalRingHomomorphism.html   (341 words)

 Ring homomorphism - Wikipedia, the free encyclopedia In abstract algebra, a ring homomorphism is a function between two rings which respects the operations of addition and multiplication. If R[X] denotes the ring of all polynomials in the variable X with coefficients in the real numbers R, and C denotes the complex numbers, then the function f : R[X] → C defined by f(p) = p(i) (substitute the imaginary unit i for the variable X in the polynomial p) is a surjective ring homomorphism. However, surjective ring homomorphisms are vastly different from epimorphisms in the category of rings. en.wikipedia.org /wiki/Ring_homomorphism   (686 words)

 Monoid ring - Wikipedia, the free encyclopedia In abstract algebra, a monoid ring is a procedure which constructs a new ring from a given ring and a monoid. The set of all these functions, together with these two operations, forms a ring, the monoid ring of R over G; it is denoted by R[G]. The kernel of this homomorphism is called the augmentation ideal and is denoted by J en.wikipedia.org /wiki/Monoid_ring   (291 words)

 Kids.Net.Au - Encyclopedia > Ring homomorphism The composition of two ring homomorphisms is a ring homomorphism; the class of all rings together with the ring homomorphisms forms a category. C defined by f(p) = p(i) (substitute the imaginary unit i for the variable X in the polynomial p) is a surjective ring homomorphism. S is a ring homomorphism between the commutative rings R and S, then f induces a ring homomorphism between the matrix rings M www.kids.net.au /encyclopedia-wiki/ri/Ring_homomorphism   (371 words)

 Articles - Ring theory   (Site not responding. Last check: 2007-10-26) In mathematics, ring theory is the study of rings, algebraic structures in which addition and multiplication are defined and have similar properties to those familiar from the integers. The study of rings originated from the theory of polynomial rings and the theory of algebraic integers. A module over a ring is an abelian group that the ring acts on as a ring of endomorphisms, very much akin to the way fields (integral domains in which every non-zero element is invertible) act on vector spaces. www.deluxea.com /articles/Ring_theory   (676 words)

 NationMaster - Encyclopedia: Monoid ring   (Site not responding. Last check: 2007-10-26) If G is a group, then it is called the group ring of R over G. The ring R can be embedded into the ring R[G] via the ring homomorphism T: R->R[G] defined by Examples of non-commutative rings are given by rings of square matrices or more generally by rings of endomorphisms of abelian groups or modules, and by monoid rings. www.nationmaster.com /encyclopedia/Monoid-ring   (488 words)

 [No title] The ring H *PGLpis the commutative A*TPGLp -algebra gen- erated by an element fi of degree 3 and the element ae of degree 2p + 2. H *G(X) from the equivariant Chow ring to the equivariant cohomology ring; this is compatible with pullbacks. The homomorphism * * * Sp * kerA PGLp ! www.math.purdue.edu /research/atopology/Vistoli/PGL_p.txt   (15156 words)

 Homomorphisms   (Site not responding. Last check: 2007-10-26) A ring homomorphism f maps the ring r into or onto the image ring s, and commutes with + and *. A ring monomorphism is 1-1, with a kernel of 0. A ring isomorphism is 1-1 and onto, essentially a relabeling of the ring elements. www.mathreference.com /ring,homo.html   (577 words)

 Ring homomorphisms and isomorphisms 2x is a group homomorphism on the additive groups but is not a ring homomorphism. is a ring homomorphism which does not map the multiplicative identity to the multiplicative identity. Note the similarity with the corresponding result for groups: the kernel of a group homomorphism is a normal subgroup. www.gap-system.org /~john/MT4517/Lectures/L7.html   (513 words)

 Ring homomorphisms and isomorphisms A ring homomorphism which is a bijection (one-one and onto) is called a ring isomorphism. The kernel of a (ring) homomorphism is the set of elements mapped to 0. The kernel of a ring homomorphism is an ideal. www-groups.dcs.st-and.ac.uk /~john/MT4517/Lectures/L7.html   (513 words)

 [No title] Rings over which all full matrices are non-zerodivisors The problem remains that Theorem 3 is not readily applicable even in the commutative case, and it would be useful to have a characterization as convenient as the Bedoya-Lewin result. Recall that for a ring R the right annihilator of a subset X of R' is the set of all y c 'R such that xy = 0 for all x c X. The left annihilator of a subset Y of 'R is defined dually. Notice that the ring R (2, 3, 2) has all the good module theoretic properties that could be desired; since every full matrix is regular, the flat modules are all spacial, and it is known that the left and right global dimension is 2. www.math.rutgers.edu /~sontag/FTP_DIR/wdicks.txt   (11071 words)

 Ring homomorphism - Definition, explanation More precisely, if R and S are rings, then a ring homomorphism is a function f : R → S such that is the smallest subring contained in S, then every ring homomorphism f : R → S induces a ring homomorphism f If f : R → S is a ring homomorphism between the commutative rings R and S, then f induces a ring homomorphism between the matrix rings M www.calsky.com /lexikon/en/txt/r/ri/ring_homomorphism.php   (398 words)

 Field (mathematics) - Wikipedia, the free encyclopedia A structure which satisfies all the properties of a field except possibly for commutativity, is today called a division ring or sometimes a skew field, but also non-commutative field is still widely used. A field is a commutative ring (F, +, *) such that 0 does not equal 1 and all elements of F except 0 have a multiplicative inverse. If F is a field, and p(X) is an irreducible polynomial in the polynomial ring F[X], then the quotient F[X]/ is a field with a subfield isomorphic to F. en.wikipedia.org /wiki/Field_(mathematics)   (1630 words)

 Integral Ring Homomorphisms Recall that s×m becomes a ring and an r algebra, and multiplication in this ring is performed per component. The induced map is an r homomorphism, an s homomorphism, an m homomorphism, and a ring homomorphism. This is of course the identity element in the ring s×m. www.mathreference.com /id-ext,morph.html   (779 words)

 [No title] Z is the ring homomorphism sending the class of a finite set S to S which is just the component belonging to the trivial subgro* *up of the character map defined in (1.2). Ainv(G) is a ring homomorphism whose kernel and the cokernel are finite. We define the augmentation ideal IG of Aho(G) to be the kernel of the ring homomorphism fflG. hopf.math.purdue.edu /Lueck/lueck_burnside0504.txt   (10461 words)

 [No title] Jus* *t as any ordinary commutative ring is an algebra over the ring Z of integers, so a c* *ommutative S-algebra is an algebra over the sphere spectrum S. The definition of commutative S-algebra is rather technical; it is the resul* *t of more than twenty years of effort, by many people. D is a map of rings, * *and i*G is the formal group obtained by extension of scalars along i. We * *remind the reader that f* is not necessarily a homomorphism of abelian groups, althoug* *h it is the case that f*(0) = 0. hopf.math.purdue.edu /Rezk/rezk-units-and-logs.txt   (17437 words)

 ipedia.com: Characteristic Article   (Site not responding. Last check: 2007-10-26) In mathematics, the characteristic of a ring R with identity element 1 R is defined to be the smallest positive integer n such that n 1 R = 0. In mathematics, the characteristic of a ring R with identity element 1 The characteristic of the ring R may be equivalently defined as the unique natural number n such that nZ is the kernel of the unique ring homomorphism from Z to R which sends 1 to 1 www.ipedia.com /characteristic.html   (524 words)

 [No title]   (Site not responding. Last check: 2007-10-26) The ring M_2(RR) is an example of an infinite non-commutative ring. Consider the ring R of continuous functions from RR to RR, with addition and multiplication are defined by (f + g)(x) = f(x) + g(x), (fg)(x) = f(x)g(x) for f, g in R, and for x in RR. We noted that this ring is a commutative ring with identity. math.smsu.edu /belshoff/532/sample_final_questions.txt   (483 words)

 ABSTRACT ALGEBRA ON LINE: Rings (part 2) A ring R in which each nonzero element is a unit is called a division ring or skew field. A ring homomorphism that is one-to-one and onto is called an isomorphism. Any ring R is isomorphic to a subring of an endomorphism ring End(A), for some abelian group A. Definition. www.math.niu.edu /~beachy/aaol/rings2.html   (1462 words)

 [No title] The ring S-1A is called a localization of A. This terminology arises from consideration of rings of continuous functions, with values, say, in a field. If A is a ring of functions on a space X and if Y (X is a subspace, which could be a single point, let S = S(Y) (A denote the subset of functions which have no zeros on Y. It is clear that S is a multiplicative set. In this situation, the localization S-1A is often denoted A(P), and referred to as the localization of A at the prime P. The ring A(P) is a local ring, that is, a ring with a unique maximal ideal. www.stanford.edu /class/math210b/LOCALIZATION.doc   (1350 words)

 ABSTRACT ALGEBRA: OnLine Study Guide, Section 5.2 for all a,b in R. A ring homomorphism that is one-to-one and onto is called an isomorphism. S. An isomorphism from the commutative ring R onto itself is called an automorphism of R. Proposition 5.2.2. The set of ordered pairs (r,s) such that r is in R and s is in S is a commutative ring under componentwise addition and multiplication. www.math.niu.edu /~beachy/abstract_algebra/study_guide/52.html   (361 words)

 Abstract algebra:Ring Homomorphisms - Wikibooks, collection of open-content textbooks In other words, f is a ring homomorphism if it preserves additive and multiplicative structure. Rings, like groups, have factor objects that are kernels of homomorphisms. Furthermore, we have an isomorphism theorem for rings analogous to the one for groups: en.wikibooks.org /wiki/Abstract_algebra:Ring_Homomorphisms   (155 words)

 Introduction   (Site not responding. Last check: 2007-10-26) Univariate polynomial rings may be defined over any ring R. Let us denote the univariate polynomial ring in indeterminate x over the coefficient ring R by P=R[x]. A ring homomorphism taking a polynomial ring R[x] as its domain requires 2 pieces of information, namely, a map (homomorphism) telling how to map the coefficient ring R, together with the image of the indeterminate x. The coefficient ring map may be omitted, in which case the coefficients are mapped into S by the unitary homomorphism sending 1_R to 1_S. www.math.uga.edu /~matthews/DOCS/MAGMA/text362.html   (473 words)

 [No title] Remark We have already seen that the spectrum of a commutative ring is a ringed space; that's the motivation for this definition. Because these homomorphisms are compatible with the restriction homomorphisms, the univeral property of the direct limit produces a homomorphism R → Spec(R) is a fully faithful contravariant functor from the category RG of commutative rings into the category SCH of schemes; the image is naturally equivalent to the category of affine schemes. odin.mdacc.tmc.edu /~krc/agathos/schem2.html   (1105 words)

 8.2 Functors of points   (Site not responding. Last check: 2007-10-26) This is an example of a functor F from finite rings to finite sets; i. To every finite ring we associate the group of units in the ring. R is a the quotient of the ring www.imsc.res.in /~kapil/crypto/notes/node40.html   (1505 words)

 Structure Operations   (Site not responding. Last check: 2007-10-26) The main structure related to a polynomial ring is its coefficient ring. Given a polynomial ring P=R[x_1,..., x_n] of rank n with coefficient ring R, together with a ring S, construct the polynomial ring Q=S[x_1,..., x_n]. In its general form a ring homomorphism taking a polynomial ring R[x_1,..., x_n] as domain requires n + 1 pieces of information, namely, a map (homomorphism) telling how to map the coefficient ring R together with the images of the n indeterminates. www.umich.edu /~gpcc/scs/magma/text593.htm   (369 words)

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