Do semirings always require neutral elements? Do semirings always require neutral elements?
In particular, an algebraic structure $(S, +, ×, 0, 1)$ is said to be a semiring if $(S,+, 0)$ and $(S,., 1)$ are monoid and multiplication distributes over addition from both left and right. But in general, some authors prefer to drop out the requirement of neutral elements. Likewise,  in wolfram mathword, semiring is defined as a commutative semigroup under addition and a semigroup under multiplication. So it seems that a semiring neither requires neutral elements nor it requires absorption law. So  would i be right to have a semiring without neutral elements? Though the similar questions might have been asked in this forum earlier also, but i couldn't clarify my confusions yet so i thought to pose this question again.

Yes, you would be right, even in the last line of the wikipedia definition of semiring it is written that some authors prefer to define semirings without 0 and 1.
This definition can be useful because in this way any subset of a field is a commutative semiring (since it does not necessarily have to have 0, 1, or inverse elements, but by maintaining the operations of the field it respects the commutative, associative and distributive property of the sum and of the product).
