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please answer ALL questions 8. Suppose R is a ring such that for all rt ER, (a + b)(a - b) = q? - 62. Prove that Ris commutative. 9. If R is a ring such that for all r e R, r2 = r, prove that every element of r is its own additive inverse. (Hint: Start with (a + a)?). 10. If R is a ring such that for all r ER, p2 = r, prove that R...
12. NEZ True] [False] A maximal ideal is prime. [True] [False] The ring Q[x]/<r? + 10x + 5) is a field [True] [False] If R is an integral domain and I c R is an ideal, then R/I is an integral domain as well [True] [False] The map : M2(Q) - Q defined by °(A) = det(A) is a ring homomorphism. [True] [False] If I, J are distinct ideals of a ring R then the quotient rings R/T and R/T...
Q1 (7 points) For k e R any constant, find the general solution to xa y" + (1 – k)x y' = 0, and use it to show that when k < 0, all solutions tend to a constant as x + 2O.
The solution of the IVP dy dx = (ax+by+ 1)2 - ; y(0)=0, where a ER and b ERVO) is Select one: a. (ax+by+1)(1 + bx)= 1 2 b. (ax+by+1)= 1-bx c. ax + by=1 d. (ax +by+1)(1 - bx)2 = 1 e. (ax+by-1)(1 - bx)= 1 f. (ax+by+1) (1 - bx)= 1 g. (ax +by+1)(1-bx)= 1 h. (ax+by+1)(1 - bx)=3
The solution of the IVP dy dx = (ax+by+112 - 8 yO=0, where a ER and b ERVO} Select one: O a. (ax + by + 1) (1 + bx)=1 O b. ax +by=1 O c.(ax +by+1) (1 - bx)=1 o d. (ax+by-1 (1-bx)= 1 2 O e. (ax+by+1)= 1-bx of. (ax +by+1)(1-bx)2 = 1 O 8. (ax +by+1)(1-bx)=1 oh. (ax +by+1)(1 - bx)=3
117. If R is any ring with identity, let J(R) denote the Jacobson radical of R. Show that if e is any idempotent of R, then J(e Re) eJ(R)e. 117. If R is any ring with identity, let J(R) denote the Jacobson radical of R. Show that if e is any idempotent of R, then J(e Re) eJ(R)e.
Let R be a ring with identity 1. Suppose that 08 a € R satisfies a? = a. Show that for each TER, there exists a positive integer n such that [(1 – a)ral" = 0. What is the smallest possible value of n that works for all r ER?
The solution of the IVP dy dx (ax +by+12-6: YO)=0, where a ER and b ERVO) is Select one: a. ax+by=1 b. (ax+by+1)(1-bx)= 3 c. (ax+by-1)(1-bx) = 1 d. (ax+by+1)(1-bx)2 = 1 e. (ax+by+1)(1 - bx)= 1 2 f. (ax+by+1) = 1- by o g. (ax +by+1) (1 - bx)=1 n. (ax +by+1)(1+bx)= 1
Correction: first problem is #2, not #1. Please show all steps in the proofs. Definitions for problems #2 through #5: Let C be the set of all Cauchy sequences of rational numbers, with the operations of addition and multiplication defined on C by (an) + (bn) = (an + bn) and (an)(bn) = (anbn). Let N be the subset of C consisting of all null sequences in c. Properties of a ring: A1. (a + b) +c= a + b...
The solution of the IVP dy dx = (ax+by+1)2 - Ø; y(D)=0, where a ER and b ERVO) is Select one: a. ax+by=1 b. (ax+by+1)(1-bx) - 3 c. (ax+by-1)(1 - bx)= 1 d. (ax +by+1)(1 - bx)2 = 1 e. (ax+by+1)(1-bx)= 1 2 f. (ax+by+1)= 1-bx g. (ax+by+1) (1 - bx)= 1 h. (ax+by+ 1)(1+bx) = 1 O O