Let R be a ring with unity, and assume I has no nonzero zoro-dinzors, Lot a, bER and полгого assume that ab = 1. show that ba= I too and therefore a and bare units.

Q: 7) Let R be a ring with 1, M a left R – Module, and N a submodule of M. Prove that if both M/N and N…

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Q: 5. Let R be a ring and r1,., r, € R. Prove that the subset (r1,., "n) = {Airı + Anrn | A1,. An E R}…

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Q: 4. Let R be a nontrivial ring. Prove that R is an integral domain if and only if the following…

A: To prove that a nontrivial ring R is an integral domain if and only if the cancellation property…

Q: 12. (a) Let R be a ring without zero divisors. Show that if ab = 1 then ba = 1 as well. Thus, a and…

A: Given that R is a ring without zero divisors. If ab=1, show that ba=1

Q: Suppose that K is a commutative ring with identity. If and I are ideals of R for which R/I≈ R/J as…

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Q: Find the final amount of money in an account if $3,900$3,900 is deposited at 7%7% interest…

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Q: Theorem 79. Let a, b, c, d ∈ R where R is a ring then (a + b)(c + d) = ac + ad + bc + bd. Note.…

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Q: Q1. a) Give an example of a semi-ring that is not a semi -algebra. b) Give an example of a…

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Q: 3. Let R be a commutative ring. Let x, y E R such that their characteristics are p. Show that (x +…

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Q: aw- Let Z[/2] = {a +b/2 |a, b e Z} and let H = { b а 2b : a,b eZ }. a Show that Z[ /2] and H are…

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Q: It is known that (ℤ,⋆,⊙) where ? ⋆ ? = ? + ? − 1 and ? ⊙ ? = ? + ? − ?? is a ring. Does it have…

A: Please check step 2 for the solution.

Q: Let R be a ring with unity. Show that {а)3D( 2 хау: х, у є R}. finite

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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: Give an example of ring elements a and b with the properties thatab = 0 but ba ≠ 0.

A: We have to give the example of ring elements that follow the following properties: ab=0 but ba≠0.

Q: O) be the smallest ideal of R that contains a. I6 Ris a commulative Ring uith unity , show That…

A: Given, a be the smallest ideal of R that contain a.If R is a commutative Ring with unity,   To prove…

Q: 1S prove that (a + b) = a² + b².

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Q: 12. Let R² be the set of all pairs of real numbers R² = {(a,b)|a, b = R}. The addition is still…

A: is the set of all pairs of real numbers The multiplication is defined by

Q: Let R be a ring with unity. Show that (a) = { E xay : x, y e R }. finite

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Q: 4. a) What are the units of Z3? b) What are the units of Z4? c) Find the units in Z3 × Z4. d) Look…

A: Step 1: Step 2: Step 3: Step 4:

Q: 8. If 1 and 2 are two ideals of the ring R, prove that the set 1₁+= {x+yx Eh, yЄh} is an ideal of R…

A: To prove that the set I1​+I2​={x+y∣x∈I1​,y∈I2​}  is an ideal of the ring (R) and contains each of…

Q: Is there a ring that has exactly one proper ideal?

A: We have to find a ring that has exactly one proper ideal.

Q: . Consider the ring Z with ideal . Give the ring tables of Z/ .

A: Consider the ring  with ideal HereLet R = We know thatAndWe have to write the addition and…

Q: Q1) In a ring Z, find the least common multiple of {2,3,8,12).

A: Given problem Given that   In a ring Z , We have to find the Least common multiple of { 2,3,8,12…

Q: If R and S are rings, define the direct sum of R and S, ROS by RS = {(r, s) : r € R, s € S} where…

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Q: Let R be a ring with unity and assume p, q, r ∈ R∗ Find (pqr)-1and also prove that it’s the inverse…

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Q: Rp { -b a | a, b = Zp}

A: It is given that,  Rp = ab-ba : a, b ∈ ℤp We have to prove that,  Rp is a commutative ring with unit…

Q: 1) Let R be a commutative ring and P,Q ideals in R. The product of the ideals P, Q is defined as т…

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Q: Show that the centre of a ring R is a sub ring of R. And also show that the centre of a division…

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Q: Let CR,t,,) be a Commutative ring ?

A: First I recall you Definition of commutative ring.  A commutative ring is ring in which…

Q: Let R be a ring with unity. Show that (a) = { £ xay: x, y e R }. finite

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Q: よn aring of (2,+) find In <97

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Q: Let a and b be elements of a ring R. Prove that the equation a+ x= b has a unique solution.

A: a and b are elements of ring R. We have to show that (a+x)=b has a unique solution.

Q: Let R be a ring and a, b in R. Show that ab is nilpotent implies that ba is nilpotent.

A: We know that An element "a" in a ring R is called nilpotent if an = 0 for some positive integer n.…