Mathematisches Institut WS 2001/02 der Universität München Set 11 Prof. Dr. B. Pareigis

Problem set for Advanced Algebra

(41) Let K be a field. Show that (K × K)- Mod is not equivalent to K- Mod.

(42) Show that ℝ- Mod ≅ ℂ̸- Mod (not equivalent).

(43) Let K be a field, B := M_{n}(K), _{K}P_{B} := K^{n} the set of row vectors, _{B}Q_{K} the set of column vectors. Find f : P ⊗_{B} Q ⟶ K and g : Q ⊗_{K} P ⟶ B, such that (K, B, P, Q, f, g) is a Morita context. Is this a strict Morita context? Determine the center of B and the set of twosided ideals of B.

(44) Determine the image of the maps f and g in the canonical Morita context (A, B, P, Q, f, g) for

(a) A := ℤ/(6) and P := ℤ/(2),

(b) A := ℤ/(4) and P := ℤ/(4) ⊕ ℤ/(2),

(c) A := ℤ/(6) and P := ℤ/(6) ⊕ ℤ/(2).

Due date: Tuesday, 15.1.2002, 16:15 in Lecture Hall 138