When are induction and conduction functors isomorphic
... arises : “if the functors Ind and Coind are isomorphic, does it follow that the ring R is strongly graded ?” A simple example (see Remark 3.3) shows that the answer to this question is negative. So we may ask this other question : “if R is a graded ring and the functors Ind and Coind are isomorphic, ...
... arises : “if the functors Ind and Coind are isomorphic, does it follow that the ring R is strongly graded ?” A simple example (see Remark 3.3) shows that the answer to this question is negative. So we may ask this other question : “if R is a graded ring and the functors Ind and Coind are isomorphic, ...
127 A GENERALIZATION OF BAIRE CATEGORY IN A
... dense in C, and A is any subset of C, then both N × A and A × N are nowhere dense in C 2 . Therefore if B is a set of first ωα -category in C, and A is any subset of C then both B × A and A × B are sets of first ωα -category in C 2 . Thus the set Q(2) = (Q × C) ∪ (C × Q) is a set of first ωα -catego ...
... dense in C, and A is any subset of C, then both N × A and A × N are nowhere dense in C 2 . Therefore if B is a set of first ωα -category in C, and A is any subset of C then both B × A and A × B are sets of first ωα -category in C 2 . Thus the set Q(2) = (Q × C) ∪ (C × Q) is a set of first ωα -catego ...
Monotone complete C*-algebras and generic dynamics
... cardinality of W is 2c ; where c = 2@0 : One of the useful properties of W is that it can sometimes be used to replace problems about factors by problems about commutative algebras. For example, let Gj be a countable group acting freely and ergodically on a commutative monotone complete algebra Aj ( ...
... cardinality of W is 2c ; where c = 2@0 : One of the useful properties of W is that it can sometimes be used to replace problems about factors by problems about commutative algebras. For example, let Gj be a countable group acting freely and ergodically on a commutative monotone complete algebra Aj ( ...
SECTION C Solving Linear Congruences
... 2 x 1 mod 5 because 2 3 6 1 mod 5 and 2 8 16 1 mod 5 . Hence integers 3 and 8 satisfy the linear congruence (*). They are the same solution and we count them as one solution not two. We obtain these solutions by trial and error by putting integers for x into the linear congrue ...
... 2 x 1 mod 5 because 2 3 6 1 mod 5 and 2 8 16 1 mod 5 . Hence integers 3 and 8 satisfy the linear congruence (*). They are the same solution and we count them as one solution not two. We obtain these solutions by trial and error by putting integers for x into the linear congrue ...
DISTANCE EDUCATION M.Sc. (Mathematics) DEGREE
... R is a commutative ring with unit element and M is an ideal of R, then prove that M is a maximal ideal of R if and only if R/M is a field. If R is a unique factorization domain then so is R[x ] . Let R be a commutative ring with unit element whose only ideals are (0) and R itself. Prove that R is a ...
... R is a commutative ring with unit element and M is an ideal of R, then prove that M is a maximal ideal of R if and only if R/M is a field. If R is a unique factorization domain then so is R[x ] . Let R be a commutative ring with unit element whose only ideals are (0) and R itself. Prove that R is a ...
Eigentheory of Cayley-Dickson algebras
... As a consequence, the expression αβx is unambiguous; we will usually simplify notation in this way. The real part Re(x) of an element x of An is defined to be 12 (x + x∗ ), while the imaginary part Im(x) is defined to be x − Re(x). The algebra An becomes a positive-definite real inner product space ...
... As a consequence, the expression αβx is unambiguous; we will usually simplify notation in this way. The real part Re(x) of an element x of An is defined to be 12 (x + x∗ ), while the imaginary part Im(x) is defined to be x − Re(x). The algebra An becomes a positive-definite real inner product space ...
The structure of Coh(P1) 1 Coherent sheaves
... It follows that si0 i1 cannot have any terms with the xi0 and xi1 exponents negative. In general, in s, there are thus no terms with more than one exponent negative. Next, consider the xi0 -exponent-negative terms in si0 and si1 . Because si0 i1 i2 = 0, these terms must be equal (recall the powers o ...
... It follows that si0 i1 cannot have any terms with the xi0 and xi1 exponents negative. In general, in s, there are thus no terms with more than one exponent negative. Next, consider the xi0 -exponent-negative terms in si0 and si1 . Because si0 i1 i2 = 0, these terms must be equal (recall the powers o ...
Contents - Harvard Mathematics Department
... R0 -algebra. We may also want to have that R is generated by R1 , quite frequently—in algebraic geometry, this implies a bunch of useful things about certain sheaves being invertible. (See [GD], volume II.2.) As one example, having R generated as R0 -algebra by R1 is equivalent to having R a graded ...
... R0 -algebra. We may also want to have that R is generated by R1 , quite frequently—in algebraic geometry, this implies a bunch of useful things about certain sheaves being invertible. (See [GD], volume II.2.) As one example, having R generated as R0 -algebra by R1 is equivalent to having R a graded ...
Definitions and Examples Definition (Group Homomorphism). A
... written additively, (ng) = n (g). Such homomorphisms are completely determined by (1), i.e., if (1) = a, (x) = (x · 1) = x (1) = xa. By Lagrange, |a| 10, and by property (3) of Theorem 10.1, |a| |1| or |a| 20. Thus |a| = 1, 5, 10, or 2. ...
... written additively, (ng) = n (g). Such homomorphisms are completely determined by (1), i.e., if (1) = a, (x) = (x · 1) = x (1) = xa. By Lagrange, |a| 10, and by property (3) of Theorem 10.1, |a| |1| or |a| 20. Thus |a| = 1, 5, 10, or 2. ...
Smoothness of Schubert varieties via patterns in root subsystems
... Let G be a semisimple simply-connected complex Lie group and B be a Borel subgroup. The generalized flag manifold G/B decomposes into a disjoint union of Schubert cells BwB/B, labeled by elements w of the corresponding Weyl group W . The Schubert varieties Xw = BwB/B are the closures of the Schubert ...
... Let G be a semisimple simply-connected complex Lie group and B be a Borel subgroup. The generalized flag manifold G/B decomposes into a disjoint union of Schubert cells BwB/B, labeled by elements w of the corresponding Weyl group W . The Schubert varieties Xw = BwB/B are the closures of the Schubert ...
