Ring Theory — DU B.Sc. (Hons.) Mathematics Previous Year Question Papers

(c) Let T: P£ (HR) + P3 (R) be a linear transformation defined by Find nullspace N(T) and range space R(T). Also verify Rank-Nullity theorem. (6,6,6) 6. (a) Let v and w be finite-dimensional vector spaces having ordered bases P and y respectively, and let T: V i W be linear. Then for each u € V, show that (b) Let V, W and Z be vector spaces and let T: V i W and U: W i Z be linear transformations. Then prove that (i) If UT is one-to-one, then T is one-to-orie. (ii) If UT is onto, then U is onto. (cHetA=[::]andletp={(:),(;)}beanordered basis for HR2. Find [LA]p and also find an invertible matrix Q such that [LA]p = Q-]AQ. where LA is a left~multiplication transformation. 1. (a) Define a nilpotent element in a riiig. Show that the nilpotent elements of a commutative ring R form aosubring of R. (b} Give an example of a commutative ring which is not an integral domain. Show that a finite integral domain is a field. (c) In a commutative ring of characteristic 2. prove that the idempotents form a subring. (6.6.6} 2. {a) Define prime ideal of a commutative ring. Let R be a commutative ring with unity and A be an ideal of R. Then, show that R/A is an integral domain if and only if A is prime ideal of R. (b)Showthat#is afield+ 3. (c) Define subring of a ring. Let R be the ring of continuous functions from R to R. Let A = {i` € R : f(0) is an even integer}. Show that A is a subring of R, but not an ideal of R. (a) Define a ring homomorphism from a ring R to a ring S. Determine all ring homomorphisms from (b) Let R be a ring with unity 1. Prove that the mapping ¢: Z i R given by S(n) = n. I is a ring homomorpliism. Hence, show that if R has characteristic n > 0, then R contains a subring isoinorphic to Zn. (c) Let Z[Ji]={a+bJi:a,bez} and H={(:3ab):a,b€Z}.showthatz[J5]andH ai-e isomorphic as rings. (6,6,6) 4. (a) (i) Let cbeacollectionofsubspaces ofavector space V and let W denote the intersection of subspaces in €. Prove that VP is a subspace ofV. (ii) Show that W = {(a„ a2, a3)- € HR3 : 2a, + a2 5. (b) (i) Let S] and S2 are arbitrary subsets of a vector space V. If St I S2, then show that span (ii) Check whether the vectors (1,0,0), {1,0,1) and (c) Define a linearly independent subset of a vector space V. Let 5 be a linearly independent subset of a vector space V, and let v be a vector in V that is not in S. Prove that S u {v} is linearly dapendent if and only if v e span (S). (a) Let V and W be vector spaces and T: V i W be a linear transformation. Then prove that the null space N(T) and the range space R(T) are subspaces off and W, respectively. (b) Let T: ER2 i R3 be defined by T(x„ x2) = (x] - X2, x„ 2x[ + x2). Let P be the standard ordered basis for R2 and y = {(1,1,0), (0,1,1), (2,2,3)} be an ordered basis for R3. Compute [T]yp.