# find additive inverse of modular arithmetic [closed]

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?

• I'm voting to close this question as off-topic because it is a question about mathematics with no connection to cryptography. – Ella Rose Jan 7 '19 at 3:09

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.