Let R be a regular ring with 1. (i) Prove that for any a E R, there exists an idempotent e ER such that Ra Re. -- (ii) Prove that for any two idempotents e, f € R, there exists an idem- potent g E R such that Re + Rf = Rg.

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Q: Show that f:Z2 -> Z2 defined by f(n) = n2-n is a ring homomarphism.

A: The element in Z2 are {0,1} Now, f(0) = 02 - 0 = 0           f(1) = 12 - 1 = 0 Hence, f maps Z2 to…

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Q: 4. Let R be a ring. (a) Let x, y, z € R. Prove that, if x | y and y | z, then x | z. (b) Prove that…

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Q: Let R be a commutative ring. Prove that HomR(R, M) and M are isomorphic R-modules

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Q: 1. Let R be a ring with the additive identity 0. Prove that for any a E R, 0- a = 0.

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Q: - Prove that, if I is an ideai of the ring Z of integer numbers then I=, for some nɛZ'U{0}

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Q: Let T be ring containing elements e, f, both # 0T, such that e + f = 1r , e² = e, f² = f , and e · f…

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Q: (b) (-a)b = −(ab) = a(−b) for all a,b € R.

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Q: 1. Let ./ be an ideal of the ring R. Show that the ring R/J is commutative if and only if xy-yx € /…

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Q: 12. Let K be a field. Define R с :) CEK}. : a, b, c E K a 0 b 0 a 00 a (i) Prove R is a commutative…

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Q: If u is finitely additive on a ring R; E, F eR show µ(E) +µ(F) = µ(EU F)+µ(EnF) %3D

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Q: Let R be a ring. Then R is Artinian iff  R satisfies the minimum condition

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Q: (0) Let R be a commutative ring with identity and I be ideal of R. Then I is primary if and only if…

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Q: If u is finitely additive on a ring R; E, F eR show p(E) +µ(F) = u(EJ F)+u(En F)

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Q: -: Define subring. Is the set S is a subring of the ring M %3D Il 2 x 2 matrices?

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Q: Let x, y, and z be an element of a Ring Prove that x(y-z) = xy - xz

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Q: 7. Aring with identity (R. +..) is commutative if: (a) Ris an integral domain () (a+ b) - a + Zab +…

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