Math 201 B Time Vampire 2 Edward Burkard
نویسنده
چکیده
Problem 1. Let R be a ring and let M be a left R-module. (a) Let I be a right ideal in R and let IM be the abelian subgroup of M generated by the elements rm, r ∈ I, m ∈M . Then R/I ⊗RM ∼= M/IM as an abelian group. (b) If I is a two-sided ideal, then the isomorphism in (a) is that of left R-modules. (c) Let R be commutative, I, J ⊂ R be ideals. Then R/I ⊗R R/J ∼= R/(I + J) as an R-module. (d) Let R be a principal ideal domain and p ∈ R be prime. Then R/(p) ⊗R R/(p) 6= 0, i > 0. On the other hand, show that the natural injective morphism α : R/(p) → R/(p), i > 1, induces a zero map 1⊗ α : R/(p)⊗R R/(p)→ R/(p)⊗R R/(p).
منابع مشابه
Math 201 B Midterm 1
Problem 1. Let C be a category with the zero object OC (that is, an object which is both universal and co-universal). In particular, for any pair of objects M and N , we have a distinguished morphism 0M,N ∈ HomC (M,N) , called the zero morphism, which is the composition of the unique morphisms M → OC and OC → N . Let M and N be objects in C and f ∈ HomC (M,N) . Recall that a kernel of f is a pa...
متن کاملInverse p-median problems with variable edge lengths
The inverse p-median problem with variable edge lengths on graphs is to modify the edge lengths at minimum total cost with respect to given modification bounds such that a prespecified set of p vertices becomes a pmedian with respect to the new edge lengths. The problem is shown to be strongly NP-hard on general graphs and weakly NP-hard on series-parallel graphs. Therefore, the special case on...
متن کاملMath 201C Homework
i=0 ai,kv i = 0. Recall that, since v is trancendental over K, the set {1, v, v, ...} is linearly independent over K. This implies that ai,k = 0 for all k > 0 (otherwise there would be a finite linear combination of powers of v which is zero, hence v would be algebraic over K). Thus each hi above is of the form hi(x) = ai,0 (a constant polynomial), namely hi ∈ K. But then (2) implies that v is ...
متن کاملEfficient computation of 2-medians in a tree network with positive/negative weights
We consider a variant of the classical two median facility location problem on a tree in which vertices are allowed to have positive or negative weights. This problem was proposed by Burkard et al. in 2000 (R.E. Burkard, E. Çela, H. Dollani, 2-medians in trees with pos/neg-weights, Discrete Appl. Math. 105 (2000) 51–71). who looked at two objectives, finding the total minimum weighted distance ...
متن کاملA note on the complexity of the transportation problem with a permutable demand vector
In this note we investigate the computational complexity of the transportation problem with a permutable demand vector, TP-PD for short. In the TP-PD, the goal is to permute the elements of the given integer demand vector b = (b1; : : : ; bn) in order to minimize the overall transportation costs. Meusel and Burkard [4] recently proved that the TP-PD is strongly NP-hard. In their NP-hardness red...
متن کامل