Answered: maximal

Math

Advanced Math

maximal

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Suppose that R₁ is a ring with addition +1 and multiplication 1, and that R2 is a ring with addition +2 and multiplication 2. Prove that the set R₁ x R2, with addition given by (11, 12) + (81, 82) := (1 +1 81, 12 +2 82) and multiplication given by (11, 12). (81, 82) := (r1·181, 12 2 82) is a ring. This ring is called the product of rings R₁ and R2.
6. Suppose that (M, +,.) be a maximal ideal of the commutative ring with identity (R, +,.) and x E M, then (a) (M, x) = R (b) x R (c) rad M = 0 (d) No Choice
Let M be a commutative ring with identity x and R be a maximal ideal of M. Then prove that R is prime ideal of M. Also show by an example that the existence of identity is necessary.
Que 409: If S be an ideal ofg a ring Z of all integers. Then S is maximal if and only if it is generated by some prime integer.
4: prove that Let R be a commutative ring with identity and I be a maximal ideal of R. Then the quotient ring R/I is a field.
Let R be a ring with unity which has exactly one maximal left ideal M. Prove that none of the elements in M has a right multiplicative inverse.
2. Given f is a homomorphism from the ring (R,+, ) onto the ring (R',+', '), prove that a) if (I,+, ) is a maximal ideal of (R,+, ·), then the triple (f(I),+', ') is a maximal ideal of (R',+', '), b) if (I', +', ') is a maximal ideal of (R',+', '), then the triple (f-'(1'),+, ·) is a maximal ideal of (R, +, :).
Let R be a commutative ring with identity. Show that every maximal ideal is also a prime ideal. Give an example to show that not every prime ideal is maximal.
Let R be a commutative ring with unity and let M be a maximal ideal of R such that M2 = {0}. Show that if N is any other maximal ideal of R, then N= M.
Suppose that R is a commutative ring with unity and that I is an ideal of R. Prove that the set of all x ϵ R such that xn ϵ I for some positive integer n is an ideal of R.
Prove directly that a maximal ideal is irreducible.
13) The integer n = a) If n = 3 (mod 9), then what is x ? b) If n = 3 (mod 11), then what is x ? Hint for #13: 12700140x3243243 is missing the digit x. Consider the number 1234. By writing each digit in scientific notation, we have 1234 1 x 10³ +2× 10² +3 × 10¹ + 4 × 10⁰ Then notice that 10 = 1 (mod 9). When we reduce mod 9, we obtain 1234 1 x 10³ +2× 10² +3 x 10¹ + 4 x 10⁰ = = 1x 1³ + 2 x 1² + 3 × 1¹ + 4 × 1⁰ = 1+ 2+ 3+ 4 = 10 = 1 (mod 9). Since 10 = -1 (mod 11), when we reduce mod 11, we obtain 1234 1 x 10³ +2× 10² +3 x 10¹ + 4 x 10⁰ = = 1x (-1)³ +2× (-1)² +3 × (-1)¹ +4× (-1)⁰ = -1 + 2-3 + 4 = 2 (mod 11).