Is there any difference between the definition of a commutative ring and field? Is a commutative ring a field? A set equipped with addition and multiplication which is abelian over those two operations and it holds distributivity of multiplication over addition?
 A: A key difference between an ordinary commutative ring and a field is that in a field, all non-zero elements must be invertible. For example:
$\Bbb{Z}$ is a commutative ring but $2$ is not invertible in there so it can't be a field, whereas $\Bbb{Q}$ is a field and every non-zero element has an inverse. 
Examples of commutative rings that are not fields:


*

*The ring of polynomials in one indeterminate over $\Bbb{Q}, \Bbb{R}$, $\Bbb{C}$, $\Bbb{F}_{11}$, $\Bbb{Q}(\sqrt{2},\sqrt{3})$ or $\Bbb{Z}$. 

*The quotient ring $\Bbb{Z}/6\Bbb{Z}$

*$\Bbb{Z}[\zeta_n]$ - elements in here are linear combinations of powers of $\zeta_n$ with coefficients in $\Bbb{Z}$ (In fact this is also a finitely generated $\Bbb{Z}$ - module)

*The direct sum of rings $\Bbb{R} \oplus \Bbb{R}$ that also has the additional structure of being a 2-dimensional $\Bbb{R}$ - algebra.

*Let $X$ be a compact Hausdorff space with more than one point. Then $C(X)$ is an example of a commutative ring, the ring of all real valued functions on $X$.

*The localisation of $\Bbb{Z}$ at the prime ideal $(5)$. The result ring, $\Bbb{Z}_{(5)}$ is the set of all 
$$\left\{\frac{a}{b} :  \text{$b$ is not a multiple of 5} \right\}$$
and is a local ring, i.e. a ring with only one maximal ideal.

*I believe when $G$ is a cyclic group, the endomorphism ring $\textrm{End}(G)$ is an example of a commutative ring.
Examples of Fields:


*

*$\Bbb{F}_{2^5}$

*$\Bbb{Q}(\zeta_n)$

*$\Bbb{R}$

*$\Bbb{C}$

*The fraction field of an integral domain

*More generally given an algebraic extension $E/F$, for any $\alpha \in E$ we have $F(\alpha)$ being a field.

*The algebraic closure $\overline{\Bbb{Q}}$ of $\Bbb{Q}$ in $\Bbb{C}$.
A: In a commutative ring not every nonzero element has a multiplicative inverse unlike the requirement in a field that every nonzero element has a multiplicative inverse.
