Struggling to understand principal ideals in Ring Theory I studied group theory last year and the concept of $&lt;g&gt;$ or $&lt;g,h&gt;$ for elements $g,h$ of a group $G$ made complete sense as it was just the set of all the products of elements within $&lt; &gt;$ and their inverses just like how span works.
Now when I get to Ring Theory (old edition with NO UNITAL ELEMENT requirement) the definition of $&lt;r&gt;$ and $&lt;r,s&gt;$ for $r,s$ in a ring $R$ just doesn’t make sense anymore as it’s no longer just the products (and addition) of the elements within $&lt; &gt;$ but instead involves elements outside of $&lt; &gt;$ as well.

I just can’t picture in my head what’s going on here and it’s making harder following my classes harder. For example one line in a proof we were doing was:

“Clearly $&lt;a&gt;^3 \subseteq RaR$“ ($R$ is a ring and $a$ is an element in $R$) but this isn’t at all clear for me.
Since you are specifying that your ring doesn't have a unit, you need to be careful when defining ideals: $\left&lt;a\right&gt;$ is not the set of all multiples of $a$, since otherwise $a$ would not necessarily be an element of $\left&lt;a\right&gt;$ as there is no unit.

$\left&lt;a\right&gt;$ is the smallest subset of $R$ that


contains $a$
is an additive subgroup 
is stable by external multiplication with elements of $R$.


This last requirement, which is the one that bothers you, is actually quite natural. You can think of how an even number is still an even number when you multiply it by any number, odd or even (here we are looking at the set of even numbers, $\left&lt;2\right&gt;$, inside $\Bbb Z$).

See this question regarding why ideals play a more important role than subrings in ring theory.
Since you are specifying that your ring doesn't have a unit, you need to be careful when defining ideals: $\left&lt;a\right&gt;$ is not the set of all multiples of $a$, since otherwise $a$ would not necessarily be an element of $\left&lt;a\right&gt;$ as there is no unit.

$\left&lt;a\right&gt;$ is the smallest subset of $R$ that


contains $a$
is an additive subgroup 
is stable by external multiplication with elements of $R$.


This last requirement, which is the one that bothers you, is actually quite natural. You can think of how an even number is still an even number when you multiply it by any number, odd or even (here we are looking at the set of even numbers, $\left&lt;2\right&gt;$, inside $\Bbb Z$).

See this question regarding why ideals play a more important role than subrings in ring theory.