(b) (6 marks) Find a decomposition of the module Z/10Z over the ring Z into a direct sum of indec...
Give an example of a non-PID over which every finitely generated module is a direct sum of cyclic modules. We do this by finding a ring R that is not an integral domain. Then use the fundamental theorem of finite abelian groups.
Please help! Thank you so much!!! 1. A module P over a ring R is said to be projective if given a diagram of R-module homomor phisms with bottom row exact (i.e. g is surjective), there exists an R-module P → A such that the following diagram commutes (ie, g。h homomorphism h: (a) Suppose that P is a projective R-module. Show that every short exact sequence 0 → ABP -0 is split exact (and hence B A P). (b) Prove...
2 Factor the following using the decomposition method: (6 marks) a z? - 3. - 70 24x + 3 -253 - 12 b. 482? C. -7°
4 -(1,5+1,5+2 marks) Explain why a) the groups z, and S, are not isomorphic b) the groups Z, x Z2 and Z, xZ, xZ, are no isomorphic; c) the function from ring R-a+b/2a,bEto ring S-abv3a,bE defined by fla+bv2abv3 is not an isomorphism. 4 -(1,5+1,5+2 marks) Explain why a) the groups z, and S, are not isomorphic b) the groups Z, x Z2 and Z, xZ, xZ, are no isomorphic; c) the function from ring R-a+b/2a,bEto ring S-abv3a,bE defined by fla+bv2abv3...
(7) Let R be a ring with 1 and let M be a unital left R-module. If I is a right ideal of R then the annihilator of I in M is defined to be AnnM(I) = {m € M: am=0 for all a € 1}. (a) Prove that Annm(I) is a submodule of M. (b) Take R = Z and M = Z/3Z Z/102 x Z/4Z. If I = 2Z describe AnnM(I) as a direct product of cyclic groups.
Problem 6 Charge Q is uniformly distributed over a circular ring on the xy plane with an inner and outer radius a and b, respectively. Calculate the electric field at any point on the z axis by using Coulomb's law. Then, calculate the electric potential on the z axis and use this expression to find the z component of the electric field. Check that the electric field calculated through the potential is the same as the one calculated by using...
Find the Laurent series (expressed as a sum) of the following functions: a) f(z) =-sinh(z) C' b)f(z) =-e Find the Laurent series (expressed as a sum) of the following functions: a) f(z) =-sinh(z) C' b)f(z) =-e
Question 6 (18 marks) (a) Given that the rate constant k for the first-order decomposition of compound X is 2.65 x 10-9 s', calculate the percentage of compound X that has decomposed in the first 2250 seconds after the reaction begins. (4 marks) (b) Consider the first order reaction: W2 → 2 Y. If [W2]=0.8 M initially and 0.17 M after 160 seconds, what will [W2] be after 350 seconds? (4 marks) (c) Data for the reaction 3A + 5B...
so Evaluate the constants A, B, and C in the following partial fraction decomposition. Z -1 f(2)={-1-)= + B Telje (Hint: To find B, take Lim 012-3147) In Z 2 de
Finite fields (a) Find all the ways to construct F16 as the quotient of a polynomial ring over F4 and construct the isomorphisms between them. (b) Find all the ways to construct F27 as the quotient of a polynomial ring over F3 and construct the isomorphisms between them.