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Is this monoid:

Addition modulo (take mode using m) on the set of Integers (Z m)={0,1,2,3,4,…..m-1}

i.e. For all a         a (+ modulo using m) e = e (+ modulo using m) a =a

here, e is an identity element
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It is a Monoid, bcoz there exist identity $e= 0$

Infact it is a group... Moreover an Abelian group.
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what is inverse element if it is group?

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@srestha

For any element a, inverse b = (m-a) mod m

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Can The Identity element be m
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@Nandkishor3939  I think idenitity element should belong to the given set, m is not in given set. Moreover identity element, if exists, is unique. here it is $0$

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yes, 0 is identity element

and inverse of any element w.r.t. 0

I mean 1 has inverse of 4

right?

that is why group
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@srestha Yes, If you take m= 5, then inverse of 1 is 4.

For any general case, inverse of 1 = m-1

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a (+ modulo using m) m = m (+ modulo using m) a =a

As m donot belong to our set so it is not e right?
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Yea Right.
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This is additive modulo and given is set of intergers.

Hence, Inverse of every element will be same negative number