Purchase Solution

Prove properties of a ring with additive identity 0.

Not what you're looking for?

Ask Custom Question

Let R be a ring with additive identity 0. Prove the following:

(a) For all a in R, a(0) = 0.
(b) a(-b)=-(ab).

NOTE: see attached word document for clearer notations.

Attachments
Purchase this Solution
Solution Summary

Properties of a ring with additive identity 0 are proven. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.

Solution Preview

First 0+0 =0, in any ring;

next, we observe that

a(0 + 0) = a0 = 0

to see this, use the distributive property on the left side to get

a0 + a0 = ...

Purchase this Solution
Free BrainMass Quizzes
Geometry - Real Life Application Problems

Understanding of how geometry applies to in real-world contexts

Multiplying Complex Numbers

This is a short quiz to check your understanding of multiplication of complex numbers in rectangular form.

Probability Quiz

Some questions on probability

Graphs and Functions

This quiz helps you easily identify a function and test your understanding of ranges, domains , function inverses and transformations.

Solving quadratic inequalities

This quiz test you on how well you are familiar with solving quadratic inequalities.

View More Free Quizzes
Related BrainMass Solutions

  • Ideals : Cyclic Module in a Commutative Ring

    im(f) is isomorphic to dom(f)/ker(f), so M is isomorphic to R/L (again noting that that f is *only* a homorphism of additive groups so this isomorphism is *only* with respect to the additive properties of modules, not the multiplicative properties

  • Sets and Rings

    Additive Associativity: For all , a+(b+c)=(a+b)+c, 2. Additive Commutativity: For all , a+b=b+a, 3. Additive identity: There exists an element such that for all , 0+a=a+0=a, 4.

  • Ring Identities

    39431 Ring Identities Equality Properties For any x,y E R (i.e. Ring R) the following equalities hold. a) 0.x=0 b) a(-b)=(-a)b=-(ab) Prove either a or b. State any properties used in your proof. Please see attached for full question.

  • Sqrarefree Integers, Fields, Conductors and Maximal Ideals

    For any positive integer f prove that the set Of = Z[fw] = {a + bfw | a, b E Z} is a subring of 0 containing the identity. Prove that [O:Of]= f (index as additive abelian groups).

  • Rings with Unity, Isomorphism, Bijectiveness and Invertibility

    Use this to prove that if u is an invertible element of a ring then ( R, +, .u ,0,uˉ¹), where a .u b =aub is a ring is ring isomorphic to R. Show also that (R, θ ,o, 1, 0), where a θ b = a+b -1, a o b= a+b -ab is ring isomorphic to R.

  • Rings

    Hence, {u, v} is a subring. We can identify u with [0] and v with [1] where Z/2Z = {[0], [1]} is the ring of integers modulo 2.

  • ring and kernel

    454796 ring and kernel i. Let R be a set with 2 laws of composition satisfying all ring axioms except the comutative law for addition. Use the distributive law to prove that the commutative law for addition holds, such that R is a ring. ii.

  • Show that a set of matrices is a ring without an identity element.

    Let 0 be the matrix with all entries 0. Then A+0=(a_ij+0)=(a_ij)=A. So R has an additive identity 0. 3. Let -A=(-a_ij), then A+(-A)=(a_ij-a_ij)=0. So -A is the additive inverse of A.

  • Rings and Subrings

    Ler R be a ring, and , prove, using axioms for a ring, the following ? The identity element of R s unique ? That -r is the unique element of R such tht (-r)+r = 0.

View More Related Solutions