I do not understand how to generate subrings. Can someone break down what a subring is and how to generate it from an element In the book we are given the following definition for a subring that is generated: The subring generated by an element is the smallest subring that must contain that element. It is every other element that must be in there because that element is in there. Recall that a subring must contain the unity if the ring contains it.
1) This definition is not clear to me. What is a subring? and what does it consist of i.e. generated elements? 
2) What is the role of unity in subrings? Unity means it contains the element 1, correct?
3) One of the problems on the homework is: Find the subring of ℝ that is generated by 1. I do not understand the necessary information. But if I had to guess I would say that this is just the integers i.e $\mathbb{Z}$  because the integers are contained inside of the set and the properties of a ring are in the integers and I can take $1$ and add or subtract as many times as I would like and can therefore generate any element in the set with $1$
 A: A sub-ring of a ring $(R,+,\times)$ is a non-empty subset $R'$ that is stable under + and x and satisfies the axioms for a ring with the binary operations + restricted to $R'$ and x restricted to $R'.$ Note that for any $g \in R$ we can define $zg$ for any integer $z$ and $g^m$ for any positive integer $m$ whether or not $R$ contains an identity element. Thus if $f \in R$ we define $$<f>=\{\sum_{i=1}^na_if^{n+1-i}|n \in Z,n>0;a_i \in Z \text { for } 1 \le i \le n\}$$ Then$<f>$is a subset of $R$ that contains $f$ and is a sub-ring of $R.$  Indeed, $<f>$ is the smallest subset of $R$ that contains $f$ and is a sub-ring of $R.$ We call $<f>$ the sub-ring of $R$ generated by $f.$ If $R$ is a ring with identity 1 these iideas must be modified in several ways. First, we can, for any integer $z$, write $z$ as a symbol for a member of $R$ that is the sum of a finite number of 1's or of -1's depending on whether $z$ is positive or negative as an integer. But we must remember that $z$ as an integer is not the same as $z$ as a member of $R.$ Then for $f \in R$ we define $$<f>=\{\sum_{i=0}^na_if^{n-i}|n \in Z,n \ge 0;a_i \in Z \text { for } 0 \le i \le n\}$$. Then $<f>$ is a ring-with-identity 1 and is the smallest ring-with-identity that contains $f.$ 
