By Nguyen Viet Dung, Franco Guerriero, Lakhdar Hammoudi, Pramod Kanwar

This quantity originated from talks given on the foreign convention on jewelry and issues held in June, 2007 at Ohio collage - Zanesville. The papers during this quantity comprise the newest ends up in present energetic study parts within the concept of earrings and modules, together with non commutative and commutative ring thought, module idea, illustration concept, and coding conception. specifically, papers during this quantity take care of subject matters similar to decomposition concept of modules, injectivity and generalizations, tilting idea, earrings and modules with chain stipulations, Leavitt course algebras, representations of finite dimensional algebras, and codes over earrings. whereas every one of these papers are unique study articles, a few are expository surveys. This e-book is acceptable for graduate scholars and researchers attracted to non commutative ring and module conception, illustration concept, and purposes

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1 0 0 0 0 f0 0 0 = Step 7. 0 2 0 0 0 f0 0 0 = 0. Step 9. 0 0 0 1 0 f0 0 0 = = 0 0 2 0 0 0 0 f0 Step 8. 2 f0 f0 0 . Step 6. 0 0 f0 0 0 0 0 1 . Step 4. 0 f0 0 0 0 0 0 f0 0 0 0 0 0 0 = 0. = 0. = 0. Thus by Step 1 through Step 9, there is a multiplication •3 on V such that •3 extends the R-module multiplication of V over R: a1 + f0 · r1 f0 · s1 2b1 c1 a2 + f0 · r2 f0 · s2 •3 2b2 c2 x y z w = , where x = a1 a2 + 2s1 r2 + 2a1 s2 + 2c1 s2 + f0 · r1 a2 + f0 · a1 r2 + f0 · r1 r2 , y = 2a1 b2 + 2r1 b2 + 2b1 c2 , z = f0 · s1 a2 + f0 · c1 s2 + f0 · s1 r2 , w = 2s1 b2 + c1 c2 .

The multiplications •1 , •2 , •3 , and •4 are well defined and they extend the R-module multiplication of V over R. Thus (V, +, •1 ), (V, +, •2 ), (V, +, •3 ), and (V, +, •4 ) are all possible compatible ring structures on V . AN EXAMPLE OF OSOFSKY AND ESSENTIAL OVERRINGS 25 13 Define θ2 : (V, +, •2 ) → (V, +, •1 ) by θ2 a + f0 · r f0 · s 2b c = a + 2r + f0 · r f0 · s 2b c . Then we see that θ2 is a ring isomorphism. Also define θ3 : (V, +, •3 ) → (V, +, •1 ) and θ4 : (V, +, •4 ) → (V, +, •1 ) by θ3 a + f0 · r f0 · s 2b c = a + 2s + f0 · r f0 · s 2b c , and a + 2r + 2s + f0 · r 2b a + f0 · r 2b = .

Note that if v = g r 0 0 because v = ve1 . By Claim 1, and noting that V = e1 V e1 +e1 V e2 +e2 V e1 +e2 V e2 , A 0 Thus e1 V e1 = f0 0 0 0 0 0 or e1 V e1 = a 0 = 0 0 0 2b 0 0 + p+f g + 0 0 + 0 0 0 c a + p + f 2b f0 0 , = g c 0 0 hence g = 0, c = 0, 2b = 0, and f0 = a + p + f . Thus a + p = 0 and f = f0 . Hence with a, b, p, c ∈ A and f, g ∈ Hom (2AA , AA ). So f0 0 0 0 = a 0 0 0 Since RR ≤ess VR , there is 0= −a + f0 0 −a + f0 0 + 0 0 0 0 −a + f0 0 with x 0 2y z ∈ R such that x 0 2y z = −ax + f0 · x 0 0 0 −2ay + 2y 0 ∈ e2 V e1 .

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