devengerQ1: what's the relation between prime and semiprime ideals.Q2: Define prime and semiprime ideals.Q3: If f and g are polynomials in R[x]. what are deg(f +g) anddeg(f.g)?Q4: Define ring. Is every subset of a ring R also a ring?Q5: Define nilpotent element. Are 1 and 3 nilpotent elements of Z,?Q6: Define the Boolean ring. Is (Z,+,.) Boolean?Q7: Define the cancelation law. Is it sațisfy in any ring?Q8: What is the difference between homomorphism and isomorphism?Q9: Define maximal ideal, prime ideal.Q10: Define principal ideal ring/ nil radical.Q11: Explain the relationship/between ideal and subring.Q12: Define the concept of field. Is Z a field?Q13: Define and give an example for (integral domain ring, zero divisor).Q14: Define subring. Is the set S =is a subring of the ring Mof all 2 x 2 matrices?Q15: Define the concept of ideal and then show that the set I ={(: ): a, b e R} is not an ideal of the ring of all 2 x 2 matrices.016: What's mean by a nilpotent element of a ring R? Find all nilpotentelements of Zg.Q17:a. Let R be a ring and I, 12 be ideals of R. Is I, UI, an ideal of R?b. True or False: The set of even integers is a commutative ring.Q18:a. Find the zero-divisors of Z,.

As per the guidelines we are supposed yo solve the first question.kindly post another questions…

Answered: Q1: what's the relation between prime…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter5: Rings, Integral Domains, And Fields

Section5.3: The Field Of Quotients Of An Integral Domain

Problem 10E: Since this section presents a method for constructing a field of quotients for an arbitrary integral...

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18. Let be a commutative ring with unity, and let be the principal ideal in . Prove that is isomorphic to .
a. For a fixed element a of a commutative ring R, prove that the set I={ar|rR} is an ideal of R. (Hint: Compare this with Example 4, and note that the element a itself may not be in this set I.) b. Give an example of a commutative ring R and an element aR such that a(a)={ar|rR}.
a. If R is a commutative ring with unity, show that the characteristic of R[ x ] is the same as the characteristic of R. b. State the characteristic of Zn[ x ]. c. State the characteristic of Z[ x ].
33. An element of a ring is called nilpotent if for some positive integer . Show that the set of all nilpotent elements in a commutative ring forms an ideal of . (This ideal is called the radical of .)
17. Suppose is a ring with positive characteristic. Prove that if is any ideal of then is a multiple of the characteristic of.
. a. Let, and . Show that and are only ideals of and hence is a maximal ideal. b. Show that is not a field. Hence Theorem is not true if the condition that is commutative is removed. Theorem 6.22 Quotient Rings That are Fields. Let be a commutative ring with unity, and let be an ideal of . Then is a field if and only if is a maximal ideal of .
15. Prove that if is an ideal in a commutative ring with unity, then is an ideal in .
Let R be a commutative ring with unity whose only ideals are {0} and R Prove that R is a field.(Hint: See Exercise 30.)
22. Let be a ring with finite number of elements. Show that the characteristic of divides .
Let I be an ideal in a ring R with unity. Prove that if I contains an element a that has a multiplicative inverse, then I=R.
Let R be a commutative ring with characteristic 2. Show that each of the following is true for all x,yR a. (x+y)2=x2+y2 b. (x+y)4=x4+y4
46. Let be a set of elements containing the unity, that satisfy all of the conditions in Definition a, except condition: Addition is commutative. Prove that condition must also hold. Definition a Definition of a Ring Suppose is a set in which a relation of equality, denoted by , and operations of addition and multiplication, denoted by and , respectively, are defined. Then is a ring (with respect to these operations) if the following conditions are satisfied: 1. is closed under addition: and imply . 2. Addition in is associative: for all in. 3. contains an additive identity: for all . 4. contains an additive inverse: For in, there exists in such that . 5. Addition in is commutative: for all in . 6. is closed under multiplication: and imply . 7. Multiplication in is associative: for all in. 8. Two distributive laws hold in: and for all in . The notation will be used interchageably with to indicate multiplication.

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