1
vote
2answers
25 views

Self inverting Rings

Would it be possible for a ring to have elements that are their own additive inverses? What I mean is, would it be possible to have a ring $K$ of mathematical objects $A$ such that: $$A+A=i,\;\forall ...
0
votes
2answers
40 views

“Self invertible” group

Let there be an Abelian group with a binary operation $\ast$ on a set $S$. Let such a group respect the following propriety: $$ (X\ast Y)\ast Y = X$$ For any $X$ and $Y$ in $S$. I realize that by ...
0
votes
0answers
23 views

multiplicative inverse in factor ring

If I need to find the multiplicative inverse of an element in some $T[x]/(m)$ factor ring, do I need to solve a diophantine equation to get the solution? Let the element be $f$. Then $fu \equiv 1$ ...
0
votes
1answer
58 views

Find the inverse of a matrix in $GL(2\,,\, \Bbb Z_{11})$.

What are the necessary steps and reasoning for calculating the following matrix in GL(2,$\Bbb Z_{11}$): $M = \begin{pmatrix} 2&6 \\3&5 \end{pmatrix}$. I found the answer to be ...
3
votes
1answer
70 views

Is there a name for an algebraic structure like this?

I'm self studying abstract algebra. I see that in rings there's no requirement for a multiplicative inverse. Is there something similar except with no requirement for an additive inverse. For ...
2
votes
0answers
44 views

Taking the (pseudo)inverse of a monoid operation.

Let $M$ be a monoid with binary operation $f : M \times M \to M$. I'm interested in functions $g : M \to M\times M$ that obey the property: $$ f(g(m)) = m $$ I want to understand what all of the ...
0
votes
0answers
88 views

Subring of the field of rational numbers

Let $R=\{a\cdot2^n\mid a,n \in \mathbb{Z}\}$ be a subring or the field of rational numbers $\mathbb Q$. i) What kind of elements are invertible in $R$? ii) Prove that $R$ is a principal ...
2
votes
1answer
107 views

Finding the multiplicative inverse of an element in $\mathbb Q[x]/(x^3-2)$

I have a problem here that asks: "Express the multiplicative inverse of $1+2^{1/3}-3\cdot2^{2/3}$ as $a_0+a_1\cdot2^{1/3}+a_2\cdot2^{2/3}$." I believe they are asking us to find it by utilizing the ...
1
vote
2answers
193 views

Multiplicative Inverses in Non-Commutative Rings

My abstract book defines inverses (units) as solutions to the equation $ax=1$ then stipulates in the definition that $xa=1=ax$, even in non-commutative rings. But I'm having trouble understanding why ...
4
votes
2answers
61 views

Proving inverses with permutations?

Prove (if f and g are permutations) that $(f \circ g)^{-1} = g^{-1} \circ f^{-1}$. My teacher gave me the hint that it has something to do with identity mapping, but that doesn't help me at all. ...
1
vote
0answers
44 views

A question about unital Algebra over a Field

Let $F$ be a field and $(K,\bullet)$ be a unital $F$-algebra. For each $v \in K$ let $S(v)$ be the set of all $a \in F$ satisfying the condition that $v -a.1_K $ does not have an inverse with respect ...
2
votes
1answer
40 views

Error in understanding the theorem about the invertibility of an element(coset) of a quotient ring

There's a theorem in Abstract Algebra which states that: An element of a quotient ring $\mathbb{Z}/\langle n \rangle$ or $\mathbb{Z_n}$ that is a coset $\overline{a}$ is invertible iff $a$ and $n$ ...
0
votes
2answers
207 views

Rings | Homomorphisms | Units

Question Show that if $f :R\rightarrow S$ is a homomorphism, and if $a$ is a unit of $R$, then $f(a)$ is a unit of $S$. Show, in fact, that $f(a^{−1}) = f(a)^{−1}$ for any unit $a$ of $R$. Attempt ...
2
votes
2answers
100 views

Conditions under which $\Bbb{R}^2$ is a field.

On my assignment one of the questions asks me to prove that $\Bbb R^2$, a set containing ordered pairs of real numbers, with the operators: $(a,b)+(c,d)=(a+c,b+d)$ $(a,b)\cdot(c,d)=(ac,bd)$ is not ...
1
vote
1answer
147 views

Terminology question; inverse vs complement in Boolean algebra

This was said at a lecture I attended: $e$ is neutral element for operation $*$ if $\forall x (x*e=x \wedge e*x = x)$. So, for example 0 is n. e. for disjunction and 1 is n. e. for ...
2
votes
3answers
126 views

Determine invertible and inverses in $(\mathbb Z_8, \ast)$

Let $\ast$ be defined in $\mathbb Z_8$ as follows: $$\begin{aligned} a \ast b = a +b+2ab\end{aligned}$$ Determine all the invertible elements in $(\mathbb Z_8, \ast)$ and determine, if possibile, ...
3
votes
1answer
172 views

Need help finding inverse under $a\circ b = a^b b^a$

I'm going through some problems in Theorems, Corollaries, Lemmas, and Methods of Proof, and I'm stuck at a certain problem that seemed very interesting until I couldn't solve it for the life of me. ...
2
votes
2answers
177 views

Commutative monoid, unsure of how to deal with negative elements, inverses and subtraction

We are working with a commutative monoid. Subtraction might be useful for us. However, we're not sure how to proceed -- negative elements have no meaning. How do we deal with allowing subtraction ...
1
vote
6answers
603 views

Proof that $\mathbb{Z}$ has no zero divisors

Everyone knows the rules of zero divisors like $$\forall \alpha,\beta\in\mathbb{R}\;:\;\alpha\cdot\beta = 0\Rightarrow\alpha=0\vee \beta=0.$$ But how can I prove it for $\mathbb{Z}$? My first try was ...