Confusion in the definition of field because of "nonzero element" statement

Question

I am newbite in abstract algebra and learning it from books by myself. I saw the following definition in my book:

Let R be a commutative ring with unity.Then, R is called a field if every nonzero element of R is a unit.

My question is that what the nonzero element of R means. Is it the number 0 ? or is it the identity element of the addition operator ? Other books also write it such as $R-{0}\$ , so I confused here. Why they use "nonzero element " term there. Can you give me more clear definition ?

If it is wanted to say the identity element of the addition operation can we say that:

Let R be a commutative ring with unity.Then, R is called a field if every element of R except for the identity element of the addition operation is unit,i.e,every element of R except for the identity element of the addition operation has inverse element with respect to multiplication operation.

Sorry for this basic question,but there is none around me to ask it, so I depend on this site as the last way.

Answer 1

Your understanding is correct.

$0$ or "zero-element" is just a short hand for "identity of the additive abelian group $(R,+)$".

Every ring must have a zero-element as a consequence of the definition. Don't think of this $0$ as the real number zero, it's more general.

For a ring with identity, $R^*$ is used to indicate the group of units. $$R^*=\{x\in R \text{ such that } \exists x^{-1}\in R\}$$

In case $R$ is a field, $R^*= R\setminus\{0\}$.

Answer 2

When talking about rings and fields, zero always refers to the additive identity, and commonly 1 may be used to refer to the multiplicative identity.

Given that rings and fields don't even need to contain the standard numbers (as integers, or reals, or whatever) this is an important distinction.