Identity elements of semirings

Question

When we define a group the identity element can be any suitable one. E.g.
$(\mathbb{N}, 0, +, -)$ or $(\mathbb{N}, 1, \cdot , N_i^{-1})$ are two groups with $2$ different identity elements.
Now it is not clear to me if in the definition of a semiring when it is stated that the identity element is $0$ and $1$ if that is meant literary. I.e. that the identity elements of the two semirings have to be $0$ and $1$.
I am asking because when reading about groups, I encountered also some definitions that used $1$ to denote the identity element but in reality they meant $I_m$ (whatever is the Identity of the set $M$).
Could someone please confirm?

Answer 1

A (semi)ring has 2 basic operations, the addition and the multiplication.

Independently, both are required to be associative and to have an identity element (so that they are monoids).

The identity element for addition is denoted by $0$, and the identity element for multiplication is denoted by $1$.

Answer 2

The identity elements $0$ and $1$ in a semiring doesn't always mean the real numbers. For example, in case of a matrix semiring $M$, additive identity $0$ means zero matrix, while multiplicative identity $1$ means identity matrix, where the operations in $M$ is usual matrix addition and multiplication. Also another example is in max-plus semiring, additive identity is $0=$-$\infty$ and multiplicative identity is real number $1=0$. Probably, there is no known semiring in which additive identity is a real number $0$ and its corresponding multiplicative identity is real $1$.