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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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    $\begingroup$ 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 '19 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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