1. D

    Zero divisors of a polynomial factor ring

    Problem As we've observed, a(x) = x^2 + x + 1 2 Z_3[x] is reducible. According to the theory (Theorem 17.5), Z3[x]=(a(x)) is not a field. Find a non-0 element q(x)+(a(x)) that has no inverse. Suggestion: Find a 0-divisor of Z3[x]=(a(x)). Attempt at a solution: I've factored x^2 + x + 1 into...
  2. A

    Polynomial Ring of a Field

    I have had some trouble with the second part of this problem. Any help would be greatly appreciated. My problem is this: If F is a field, then (x) is a maximal ideal in F[x], but it is not the only maximal ideal. Now, I have proven (I think), that F[x]/(x) \cong F and thus (x) is maximal, but...
  3. M

    How many zero divisors in a ring

    If I want to know how many zero divisors there are in the ring \mathbb{Z}_5[x]/(x^3-2), how do I go about solving that? I've noted that for p(x) = x^3-2, p(3) = 0 \space \space (\mathrm{mod} 5) and p(5) = 0 \space \space (\mathrm{mod} 5). Does that make 3 and 5 zero divisors? If I could write...
  4. G

    About ring.

    i want to ask, is every local rings is Noetherian (or every Noetherian is local ring)? thank you.
  5. L


    Hi folks, I have one problem I don't know how to solve. Find factor ring R/I and prove if ideal I is prime and/or maximal in R, if: R=C[0,1] and I={f∈C[0,1] : f(1/2)=f(1/3)=0}.
  6. J

    Prove in a Boolean Ring every Prime Ideal is a Maximal Ideal

    Let $R$ be a Boolean ring i.e. $x^2=x$ Let $P$ be a prime ideal such that it is not maximal and $M$ be a maximal Ideal such that $P \subsetneq M \subsetneq R$ Further let $y \in M\backslash P$. Now we know $y^2=y \implies y^2 - y = 0$ now let $z$ be an arbitrary element in $R$. Then $y^2z-yz=...
  7. S

    inverse of polynomial in division ring

    Hello, I'm supposed to find inverse to the polynomial: $ x^2+1+(f)$ in $Q[x]/((f),+,*)$ and $f=x^3+5$. I know the solution somehow involves extended Euclidean algorithm, but generally don't know what to do. I tried to find the coefficients for Bezout identity which worked, but don't know how to...
  8. D

    Question about Units in the polynomial ring

    Hello, I have two problems that relate, and I have been tossing them over for a couple of days but they seem to contradict each other. The first: Show that 1-t is a unit in Q[[t]]. The second: Classify all units of Q[[t]]. Now the second seems to be that all units are all non-zero zero...
  9. L

    The ring M2(Z3) - how to find the multiplicative inverses?

    Hello! I have just one question! We have the ring M_{2}(\mathbb{Z}_{3}), which are all 2x2 matrices over the field \mathbb{Z}_{3}. We need to find all the elements of ring K^{*} (the set of multiplicative inverses). I know that A \in K^{*} if the det(A) \neq 0 . But here in \mathbb{Z}_{3} we...
  10. L

    Multiplicative elements in ring M<sub>2</sub>(Z<sub>3</sub>) ...

    Multiplicative elements in ring M2(Z3) ... Hello! I have just one question! We have the ring K = M_{2}(\mathbb{Z}_{3}) , which are all 2x2 matrices over the field \mathbb{Z}_{3} . We need to find all the elements of ring K* (the set of multiplicative inverses). I know that A € K* if the...
  11. A

    Abstract Alegbra - Ring Homomorphism

    Consider the ring R = {0,2,4,6,8,10,12} with the operations being addition and multiplication mod 14. (a) Show that R is a ring with identity by finding the multiplicative identity (and prove that it’s the identity by computation). (b) Find the multiplicative inverse of 10 in R. Would I be...
  12. S

    Intersection of any collection of rings of subsets of X is a a ring of subsets of X

    This question is actually from a measure theory course but since it involves rings I posted it here. Prove that the intersection of any collection of rings (or algebras) of subsets of X is again a ring (or algebra) of subsets of X. If we look at the ring case. I know that to prove a ring we...
  13. P

    SOLVED D = R [x] \ I is not empty, it contains 0, contains no divisors of 0, and is closed

    R is an ideal. Consider polimonios ring R [x] and the principal ideal I = xR [x] R [x] generated by x Show all qe D = R [x] \ I is not empty, it contains 0, contains no divisors of 0, and is closed under multiplication. ( Show that the principal ideal x (D ^ (-1) R [x]) of D ^ (-1) R [x]...
  14. Paze

    Limit property doesn't seem to ring true

    Taking the limit as x approaches infinity of: \frac{x^2}{x-1}-\frac{x^2}{x+1} can be rewritten as: The limit as x approaches infinity of: \frac{x^2}{x-1} MINUS the limit as x approaches infinity of: \frac{x^2}{x+1} But this isn't true. If I re-arrange algebraically in the first example, I...
  15. Bernhard

    Rings of the form R[X] - Ring Adjunction

    I am reading R.Y Sharp's book: "Steps in Commutative Algebra". On page 6 in 1.11 Lemma, we have the following: [see attachment] "Let S be a subring of the ring R, and let \Gamma be a subset of R. Then S[ \Gamma ] is defined as the intersection of all subrings of R which contain S and...
  16. J

    Help with this proposition,Let A be a ring, If q is p-primary, x an element of ring..

    Hello. i am readin commutative algebra Atihyaha, i am trying to prove this statements, Let A be a ring, and q is P-primary, then if x is not element of q , we have (q:x)={y | yx is in q} is P-primary. I need your help . THANK YOU
  17. B

    Principal Ideal Ring

    Let X be a set then (\matchal{P}(X),\Delta,\cap) forms a commutative ring with unity X. Although not a principal ideal domain if X has at least two members, I was wondering if it was necessarily a principal ideal ring. Namely if I was an ideal whether I=(\bigcup I) necessarily. I think it's...
  18. S

    Ring theory

    In a regular ring R every principle ideal is generated by an idempotent element. It is given that R is commutative with identity element.
  19. J

    Let A be a ring, prove that if AB=I and BA=I .....

    Can you help me prove this statements. Let A be a ring, and A, B be 2 matrixs. if $A_{mxn}B_{nxm}=I_m$ and $B_{nxm}A_{mxn}=I_n$ then m=n. Im and In are identity marix. Thank you .
  20. A

    Ring of integers

    Hello, let K=\mathbb Q(\sqrt d) be a number field. It is well known, that the ring of integers \mathcal O_K of K is given by \mathbb Z[\xi] with \xi = \frac{1}{2}+\frac{1}{2}\sqrt d \ \ \quad \text{ if } \quad d \equiv 1 \text{ mod }4 \xi = \sqrt d \quad \quad \ \ \quad \text{ if }...