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For a set to be called as a ring, it should have the following properties

  1. closed
  2. commutative
  3. associative
  4. Identity existence
  5. Inverse existence

but how is Z7 a ring, as there aren't any inverse element for addition as -n is not an element of z7={0,1,2,3,4,5,6}

if there are additive inverse for z7 , how to find it?

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closed as off-topic by Ella Rose Jan 7 at 3:09

  • This question does not appear to be about cryptography within the scope defined in the help center.
If this question can be reworded to fit the rules in the help center, please edit the question.

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    $\begingroup$ I'm voting to close this question as off-topic because it is a question about mathematics with no connection to cryptography. $\endgroup$ – Ella Rose Jan 7 at 3:09
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The additive inverse of $x$ is defined to be the value $y$ such the group operation $x + y$ results in 0.

If we examine $\mathbb{Z}/7$, we find that every element there does have an additive inverse. For example, for the element 2, we find the additive inverse to be 5, as $2 + 5 = 0$ (computed modulo 7).

You might be interested to learn that multiplicative inverses also exist for all elements other than 0; even though one might naïvely thing that 2 does not have an inverse (because 0.5 is not within $\mathbb{Z}/7$), we find that $2 \times 4 = 1$ (computed modulo 7), and so 4 is the multiplicative inverse of 2.

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