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Is the following set $R$ with the indicated binary operations are commutative ring with unity?

$R$ is the set of integers with the usual addition and multiplication defined $m \star n=m^{3n}$

I am studying RING THEORY right now and countered with this question. I know there are two operations in ring - usual addition and multiplication. But if such a specific binary operation is mentioned, can you please tell me how it works ?? and please correct me if I have any wrong concept.

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  • $\begingroup$ You have to check if the properties defined for a ring hold. They can be round in wikipedia, where the operation $\cdot$ in wikipedia corresponeds to $*$ in your post $\endgroup$
    – miracle173
    Jan 23, 2021 at 9:15

3 Answers 3

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This is not a ring because in a ring multiplication is distributive with respect to addition, meaning that: $(b + c) * a = (b * a) + (c * a) ,\forall a,b,c \in R$

For example, take $a=3,b=4,c=6$

We have $(4+6)*3=(4+6)^9=10^9\neq4*3+6*3=4^9+6^9=10339840$

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  • $\begingroup$ ooh, thank you I was just asking what star stands for in the question actually, thanks for your answer anyway. $\endgroup$
    – ARIGHNA
    Jan 23, 2021 at 8:37
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The definition of a ring does not require specific symbols to be used but the conventions of familiar examples, e.g. the integers, are usually adopted. So, addition is almost always denoted by $+$. Multiplication varies a bit, as it does for integers, $\times$ is sometimes used but also $.$ or just juxtaposition (nothing).

A common case for using an alternative symbol is when you want to use a familiar set but with an unfamiliar operation, especially if the definition of the non-standard operation depends on the standard one. This is your case. Your example has the usual addition but not the usual multiplication. So, a different symbol has been used partly as a reminder that it is not the familiar operation but also to allow the definition of the non-standard operation to refer to the familiar one.

As Eminem says, in this case, it fails to be a ring. I guess that this is the point of the exercise: for you to check.

You may know some group theory. The operation is most often written in a multiplication style but sometimes other symbols are used. When the group is abelian (commutative), an additive style is common. If you forget the multiplication of a ring then you have an abelian group with its operation written as addition.

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    $\begingroup$ thanks for your explanation $\endgroup$
    – ARIGHNA
    Jan 23, 2021 at 9:12
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    $\begingroup$ I'm curious about the down vote. What's the problem? $\endgroup$
    – badjohn
    Jan 24, 2021 at 19:45
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If $1$ is the unit element, then $m\star 1 = 1\star m$. But $m\star 1 = m^{3\cdot 1}$ and $1\star m= 1^{3\cdot m}.$

It's not clear what e.g. $m^{3\cdot 1}$ means. The 3-fold of $n$ is $3\cdot n =n+n+n$. But $1$ could be the unit matrix in a matrix ring. Then $m^{3\cdot 1}$ makes no sense.

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  • $\begingroup$ It is mentioned that $R$ is the set of integers. $\endgroup$
    – Christoph
    Jan 24, 2021 at 11:02

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