Let u be a unit in a ring R. Show that u divides x, for all x in R. (that is, show that x uy for some y in R).

Q: Let R be a ring and consider R x Z = { (r,n) | r e R, ne Z} Define (r,n) + (s,m) =(rts, n+ m) (r,…

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Q: Let ø : R → S be a ring homomorphism, and define ø* : R[x] → S[r] as follows. For f(r) = ana" + ...+…

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Q: Complete the proof of Theorem (the ring of cosests) by proving the right distributive law in R/I.

A: Let I be an ideal of the ring R. Then the set RI of additive cosets r+I of I in R forms a ring with…

Q: Let S be a ring. Determine whether S is commutative if it has the following property: whenever æy =…

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Q: In the ring Z, [x]. Show that 1+2x is unit. a

A: In a polynomial ring Rx the polynomial p(x)=a0+a1x+a2x2+...+anxn is unit if a0 is unit and remaining…

Q: Let R be a ring with identity 1 and let a be an element of R such that a2 = 1. Let S = { ara : r e…

A: we will use sub ring test.

Q: Let (R, +,.) be any ring. Define R × Z = {(r,n):r E R ,n E Z} and %3D (r, n) + (s, m) = (r + s,m +…

A: In the Structure T=ℝ×ℤ additional multiplication is defined as…

Q: Let (R, +,.) be any ring. Define R x Z = {(r,n):r ER ,n E Z} and (r,n) + (s, m) = (r + s,m + n) (r,…

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Q: If S is a subring of a ring R, then S[a] is a subring of R[x]. Exercise 2.35.1 Prove this assertion!…

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Q: Let R be a ring such that a² = a for all a E R, and assume that R has an identity. Show that the…

A: Given R be a ring with unity 1 such that a2=a, for all a∈R. Let b∈R be a unit in R. Therefore b-1…

Q: Let R=Z and R'= set of all even integers. Then %3D (R', +, *) is a ring, where a* b= ab V a, be R'.…

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Q: If is a homomorphism from the ring R to the ring R' , show that; a) (0)=0 b) (−r)= −(r)for all…

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Q: Let I and J be ideals of a ring R. Prove or disprove (by counterexample) that the following are…

A: Given that I and J are two ideals of a ring R Ideal Test: A nonempty subset A of a ring R is an…

Q: The ring Z is isomorphic to the ring 3Z False True

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Q: The measure u is monotone on the ring. So that µ(A) < µ(B) if ACB

A: Given that The measure is monotone ob the ring,  So we need to consider the following;

Q: = Let I andJ be two ideals of a commutative ring R. Show that: S= {r€R: ri E J Vi EI } is an ideal…

A: Ideal: A subset of a ring is called ideal of if  1. 2. Given: are ideals in a ring and . To…

Q: Let R be a ring and consider RxZ={(r,n)|r e R, n e Z}. Define (r,n) + (s,m) = (rts, n+m) (r,n)(s,m)…

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Q: If ø is a ring homomorphism from R to S. Then i. ii. Prove that (kero) is an ideal of S. Prove that…

A: Given φ is a ring homomorphism from R to S. To prove: φkerφ is an ideal of S. Given, φ: R→S is a…

Q: Let R be a ring. Prove that the set S = x R / xa = ax, a Ris a subring of R .

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Q: If f is a ring homomorphism from Z„ to Z„ such that f (1) = b, then b4*+2 = b*. O True False

A: Given f : Zm→Zn is a ring homomorphism such that f1=b Here, we have to check whether b4k+2=bk is…

Q: 37. Explain why in any ring R and any r, y, z E R: (a) 0+x= 0 (b) 1* =I (c) 0+1 = r. (d) r(y + z) =…

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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)

A: Here, we need to write the union of E and F as union of disjoint subsets then use the properties of…

Q: a) Let Ø: R – S be a homomorphism of rings R and S and let B be an ideal of S. Show that 01(B) =…

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Q: Let E = {a + bi : a, b ∈ Z, b is even}. 1. A subring S of a ring R is called an ideal of R if sr,…

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Q: a Let S= { : a,b e R}. Show that : C→ S defined by %3D a b is a ring homomorphism. a $(a + bi) = -b…

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Q: Let (R,+,.) be the ring, then R/R = {0} .a %3D R\= R .b R\R = {0} C %3D R\{0}=R •d

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

A: Solution. Since ab is nilpotent, (ab)n = 0 for some n ∈ N.

Q: Prove that if u is a unit in a ring R, then -u is a unit in R.

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Q: Q2: Let X be anon-empty set, prove that (p(X), An) is ring?

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Q: Let R be a ring and assume a∈R is not a zero divisor.Prove that if ba=ca,  then b=c.

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Q: The ring Z is isomorphic to the ring 3Z False O True

A: Z=···,-3,-2,-1,0,1,2,3,···3Z=···,-9,-6,-3,0,3,6,9,··· As the ring Z has the unity element 1 such…

Q: Let f : R S be a homomorphism of rings, 1. If K is a subringof R, Is o(K) a subring of S? If so,…

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Q: Let R be a ring such that for each a eR there exists XE R such that a'x = a. Prove the following :…

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Q: Let Z[V2] = {a + bV21 a, b E Z} and {: а 2b a, bez. H = a. Show that Z[V2] and H are isomorphic as…

A: Given are rings

Q: 3. Let R be a ring and b E R be a fixed element. Let and prove that T is a subring of R T = {rb | r…

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

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Q: The characteristic of the ring Zs x Z20 8 O 160 40 20

A: Characteristic of the ring Zm*Zn  is the LCM(m, n).

Q: a) Let R be a ring Ei a3 = a #aER %3D Prove that R is commutatve.

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: In the ring Zg [x] . Show that 1+2x is a unit.

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Q: 32. Let R be a ring. Define the center of R to be Z(R) = {a E R: ar = ra for all r E R}. Prove that…

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Q: Prove that if u is a unit in a ring R, then u is a unit in R. -

A: According to the given information, It is required to prove that if u is a unit in a ring R then -u…

Q: Let R be a commutative ring with unity and r ∈ R. Prove that if ⟨r⟩ = R, then r is a unit. Consider…

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Q: a Let S = { : a,b e R } . Show that Ø : C→ S defined by a b is a ring homomorphism. a Ф(а + bi)- a

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Q: Let R be a commutative unital ring. Let I be an ideal of R. Denote by (I) the ideal in R[a]…

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Q: Let R be a commutative ring such that a^2 = a for all a ∈ R, then show that a+a = 0.

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Q: Let R be a ring. Define a functione: R[x] ×RR by the rule e(anx" +.+ax+a0,r) = anr" + +ar+ao. In…

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Q: Let R be a ring such that a6 - = x for all æ E R. Prove that R is commutative.

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Q: Let u be a unit in ring R. Show that u divides x, for all x in R. (That is, show that x = uy for…

A: Let u be unit in a ring R  Then there exist v in R such that  uv = vu = 1  Where 1 is unity of ring…

Q: P(X), where Is (P(X), An) is A= ring (A-B) ULB-A) a where

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Continuous Probability Distributions

Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.

Normal Distribution

Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!