Field theory and distributive law.

Question

How do any subset of a Field inherit the properties-commutativity,associativity and specially Distributive law over the same operationsas that of field?Is there any intuitive way to understand that how do a subset of a field automatically acquire distributive law over the same operations as of the field?

Answer 1

It is not true that any subset of a field inherits those properties. For example if we take our field to be $K = \mathbb{Q}$ with the usual operations, then the set $\{6\}$ does not inherit the properties the operations on $K$ because it is not even closed under those operations.

However, if you require that your subset $R$ be a ring under the operations of $K$, then all the properties of the operations do descend to the operation on $R$. I think the most intuitive way to think about this is to think about the operations on $R$ as being the operations on $K$ with inputs restricted to $R$. Since the properties you mentioned (commutativity, associativity, etc.) are statements about the operations on $K$ for any possible inputs, they still hold when you restrict the inputs to $R$.