a 2b Let Z[√√√2] = {a+b√2 \a, beZ} and let H = { [ b Show that Z[√2] and Hare isomorphic as rings. a : a,b €Z}.

Q: Let R be a ring with unity 1 and char (R) = 4. %3D Then R contains a subring isomorphic to Q ZO Z3 O

A: IN the given question,  Given that: R is a ring with unity 1 and char(R)=4. we have to find: we have…

Q: 7. Suppose that (R,+,.) be a commutative ring with identity and (I,+,.) be an ideal of R. If I is…

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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: Iffis a ring homomorphism from Zm to Zn such that f (1) = b, then bak+2 = bk. True O False

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Q: Suppose that R and S are commutative rings with unities. Let ø be a ring homomorphism from R onto S…

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Q: Q:Let S and Szare two subrings of a ring (R, +..), prove that S, US2 subring of R iff either S, C S…

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Q: Let R be a ring with unity 1 and char (R) = 4. Then R contains a subring isomorphic to

A: Let R be a ring with unity 1 and char(R)=4.Then R contains a subring isomorphic to________

Q: 4. Let y: R→ S be a ring homomorphism. Prove that ': R[X] → S[X] given by soʻ (ao + a1X + ..a,X") =…

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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: If R is a commutative ring with unity and a e R, then (a) = {ar : reR}=aR.

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Q: 7. Suppose that (R, +..) be a commutative ring with identity and (I, +,.) be an ideal of R. If I is…

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Q: Consider the ring R = {r,s,t} whose addition and multiplications tables are given below. Then t.s =

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Q: Show that a ring is commutative if it has the property that ab = caimplies b = c when a ≠ 0.

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Q: Suppose that R is a commutative ring and |R| = 30. If I is an idealof R and |I| = 10, prove that I…

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Q: Iffis a ring homomorphism from Zm to Zn such thatf(1) = b, then bak+2 = bk. True False

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Q: consider the mapping from M2(z) into Z,prove or disprove that this is a ring homomorphism

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Q: Any ring R is a Jacobsen radical ring, if it is a simple ring. O مبا O

A: JACOBSON RADICAL:- The radical of the base ring R is called its Jacobson radical and denoted by…

Q: 9. Suppose that (R,+,.) be a commutative ring with identity and x E rad R, then (a) (x) = R (b) 1 —…

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Q: 4 If R is a commutativ ring, show that the characteristic of R[X] is the same the characteristic of…

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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: a Let S= { 9- a,b e R }. Show that Ø: C→ S defined by 1. a a P(a + bi) =| - b is a ring…

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Q: we give ring R, a multiplicative subset. uçR, and an Ramodule M, Give a description of the lo…

A: Given - We give a ring R, a multiplicative subset u ⊆ R, and an R-module M.  To find - Give a…

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Q: 2. Let R be a ring: The center of R is the set 3XER: ax= xa vae R? Prove that the center of a ring…

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Q: If R is a commutative ring with unity and a e R, then (a) = {ar:reR}=aR. %3D %3D

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Q: Q::Let S1 and S2are two subrings of a ring (R, +,.), prove that S, U S2 is subring of R iff either…

A: 1 Let S1 and S2be two subrings of R,+,.. First suppose that either  S1⊆S2 or S2⊆S1 we will prove…

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A: Given:- Let R,S be rings with unity ϕ:R→S a ring homomorphism. To Prove ϕ(1R)=1S

Q: * If K is a simple ring, then K a Jacobsen radical ring. T F

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Q: Q1: Let S, and Szare two subrings of a ring (R, +,.), prove that S, USz is subring of R iff either…

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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: If Ris a ring with identity and a is a unit, prove that the equation ax = has a unique solution in…

A: Let R be a ring with identity and a∈R be a unit. Prove that the equation ax=b has a unique solution…

Q: Q: Let S, and Szare two subrings of a ring (R, +,.), prove that S, US2 is subring of R iff either S,…

A: By supposinɡ S1 and S2 as two subrinɡs of rinɡ (R, +, .) To prove that S1∪ S2 is subrinɡ of R if and…

Q: 1. let S: :1 xg*Z a) Show that is a ning. ning. b) Show that ¢: S Z olefined by $ ( [Š 87)-* is a…

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Q: 9. Suppose that (R,+,.) be a commutative ring with identity and x E rad R, then (a) (x) = R (b) 1- x…

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Q: . If A, B and Ç are ideals of a ring R, prove that A (B+ C) = AB+AC.

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Q: Let R be a commutative ring with identity, and let a, b E R. Assume ab is a unit in R. Do a and b…

A: Here given R is a commutative ring with identity. and let a,b∈R and assume ab is a unit. we have to…

Q: Q: Let S, and Szare two subrings of a ring (R, +,.), prove that S, USz is subring of R iff either S,…

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Q: '. Suppose that (R, +,.) be a commutative ring with identity and (I, +,.) be an ideal of R. If I is…

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Q: Let A, B,C be rings. Let & be a ring homomorphism from A into B and ß be a ring homomorphism from B…

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Q: Le 'R' be a ring with 1.If 'a' is milpotent ele ment of 'R', Prove thał 1-a and 1 ta are Units.

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

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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: Suppose that R and S are isomorphic rings. Prove that R[r] = S[r].

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Q: Consider the ring R = [r, s,t) whose addition and multiplications tables are given below, Then ts S…

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Q: If Ø: R → S is a ring isomorphism. The Ø preserves: Units Idempotent elements

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

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Addition Rule of Probability

It simply refers to the likelihood of an event taking place whenever the occurrence of an event is uncertain. The probability of a single event can be calculated by dividing the number of successful trials of that event by the total number of trials.

Expected Value

When a large number of trials are performed for any random variable ‘X’, the predicted result is most likely the mean of all the outcomes for the random variable and it is known as expected value also known as expectation. The expected value, also known as the expectation, is denoted by: E(X).

Probability Distributions

Understanding probability is necessary to know the probability distributions. In statistics, probability is how the uncertainty of an event is measured. This event can be anything. The most common examples include tossing a coin, rolling a die, or choosing a card. Each of these events has multiple possibilities. Every such possibility is measured with the help of probability. To be more precise, the probability is used for calculating the occurrence of events that may or may not happen. Probability does not give sure results. Unless the probability of any event is 1, the different outcomes may or may not happen in real life, regardless of how less or how more their probability is.

Basic Probability

The simple definition of probability it is a chance of the occurrence of an event. It is defined in numerical form and the probability value is between 0 to 1. The probability value 0 indicates that there is no chance of that event occurring and the probability value 1 indicates that the event will occur. Sum of the probability value must be 1. The probability value is never a negative number. If it happens, then recheck the calculation.