Let R[:] be the set of all expressions a = ao+ajx+a2x² + …=La;x i=0 where a¡ e R for all nonnegative integers i. Informally, an element of R[t] is like a polynomial except that it can have infinitely many terms. (a) Carefully write down definitions of addition and multiplication operations for R[[:], analogous to the definitions for R[x] in the notes. Given a,b E R[:], your defi- nitions should indicate what each coefficient of the sum a + b and product ab is. (b) Let f = ao+ax+.+a„x" be a polynomial. I can treat f as an element of R[t] by defining a,+1,an+2,….. all to equal 0. This shows that R[x] C R[t]. If you had already proved that R[x]] was a ring, how could you use this fact to help you prove R[x] is a ring? (c) Let a E R[l]] with ao # 0. Prove that a has a multiplicative inverse in R[[x]. You may assume that the multiplicative identity element in R[x] is IR] = 1+0x+ 0x²+0x³ + •…·, and that multiplication in R[x] is commutative. [Hint. If ab = 1R1], equate coefficients and solve for bo, bị, b3, …. in turn.]
Let R[:] be the set of all expressions a = ao+ajx+a2x² + …=La;x i=0 where a¡ e R for all nonnegative integers i. Informally, an element of R[t] is like a polynomial except that it can have infinitely many terms. (a) Carefully write down definitions of addition and multiplication operations for R[[:], analogous to the definitions for R[x] in the notes. Given a,b E R[:], your defi- nitions should indicate what each coefficient of the sum a + b and product ab is. (b) Let f = ao+ax+.+a„x" be a polynomial. I can treat f as an element of R[t] by defining a,+1,an+2,….. all to equal 0. This shows that R[x] C R[t]. If you had already proved that R[x]] was a ring, how could you use this fact to help you prove R[x] is a ring? (c) Let a E R[l]] with ao # 0. Prove that a has a multiplicative inverse in R[[x]. You may assume that the multiplicative identity element in R[x] is IR] = 1+0x+ 0x²+0x³ + •…·, and that multiplication in R[x] is commutative. [Hint. If ab = 1R1], equate coefficients and solve for bo, bị, b3, …. in turn.]
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter1: Fundamentals
Section1.7: Relations
Problem 21E: 21. A relation on a nonempty set is called irreflexive if for all. Which of the relations in...
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![Let R[x] be the set of all expressions
a = ao+ajx+azxr²+.
where a; € R for all nonnegative integers i. Informally, an element of R[t] is like a
polynomial except that it can have infinitely many terms.
(a) Carefully write down definitions of addition and multiplication operations for R[*],
analogous to the definitions for R[x] in the notes. Given a,b E R[x], your defi-
nitions should indicate what each coefficient of the sum a +b and product ab is.
(b) Let f = ao+ajx++a„x" be a polynomial. I can treat f as an element of R[]
by defining an+1,4n+2;... all to equal 0. This shows that R[t] C R[:].
If you had already proved that R[t] was a ring, how could you use this fact to
help you prove R[x] is a ring?
(c) Let a E R[x] with ao #0. Prove that a has a multiplicative inverse in R[x]. You
may assume that the multiplicative identity element in R[x] is
IRH = 1+0x+ 0x² +Qx° + •…•,
and that multiplication in R[x] is commutative.
[Hint. If ab = 1R1g, equate coefficients and solve for bo, b1, b2,... in turn.]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F283b3a0d-0821-472c-a1ce-fcf162c3f2c6%2Fe66c9423-cebe-4fd5-9dcf-260b653a5d04%2F2dicff_processed.png&w=3840&q=75)
Transcribed Image Text:Let R[x] be the set of all expressions
a = ao+ajx+azxr²+.
where a; € R for all nonnegative integers i. Informally, an element of R[t] is like a
polynomial except that it can have infinitely many terms.
(a) Carefully write down definitions of addition and multiplication operations for R[*],
analogous to the definitions for R[x] in the notes. Given a,b E R[x], your defi-
nitions should indicate what each coefficient of the sum a +b and product ab is.
(b) Let f = ao+ajx++a„x" be a polynomial. I can treat f as an element of R[]
by defining an+1,4n+2;... all to equal 0. This shows that R[t] C R[:].
If you had already proved that R[t] was a ring, how could you use this fact to
help you prove R[x] is a ring?
(c) Let a E R[x] with ao #0. Prove that a has a multiplicative inverse in R[x]. You
may assume that the multiplicative identity element in R[x] is
IRH = 1+0x+ 0x² +Qx° + •…•,
and that multiplication in R[x] is commutative.
[Hint. If ab = 1R1g, equate coefficients and solve for bo, b1, b2,... in turn.]
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