To submit Let R[r] be the set of all expressions a = do +ajx+ax +.= i=0 where a¡ E R for all nonnegative integers i. Informally, an element of Rx] is like a polynomial except that it can have infinitely many terms. (a) Carefully write down definitions of addition and multiplication operations for R[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.

Advanced Engineering Mathematics
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Chapter2: Second-order Linear Odes
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please send handwritten solution for part a
To submit Let R[x] be the set of all expressions
a = do+ajx+azx +=
i=0
where a¡ E R for all nonnegative integers i. Informally, an element of R[x] is like a
polynomial except that it can have infinitely many terms.
(a) Carefully write down definitions of addition and multiplication operations for R[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[x]
by defining an+1,ɑn+2;+.. all to equal 0. This shows that R[x] CR[r].
If you had already proved that R[x] was a ring, how could you use this fact to
help you prove RÊ] 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[r] is
1gL] =1+0x+Ox +0x° + • · · ,
and that multiplication in R[x] is commutative.
[Hint. If ab = lRL], equate coefficients and solve for bo,b,b2,--. in turn.]
Transcribed Image Text:To submit Let R[x] be the set of all expressions a = do+ajx+azx += i=0 where a¡ E R for all nonnegative integers i. Informally, an element of R[x] is like a polynomial except that it can have infinitely many terms. (a) Carefully write down definitions of addition and multiplication operations for R[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[x] by defining an+1,ɑn+2;+.. all to equal 0. This shows that R[x] CR[r]. If you had already proved that R[x] was a ring, how could you use this fact to help you prove RÊ] 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[r] is 1gL] =1+0x+Ox +0x° + • · · , and that multiplication in R[x] is commutative. [Hint. If ab = lRL], equate coefficients and solve for bo,b,b2,--. in turn.]
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