Let H be a subgroup of R under +. If x*H is in H for every x in R, H is a left ideal. Since x might be drawn from H, H is closed under multiplication. If H contains a unit v, H includes xuv for every x, and H = R. Thus an ideal is proper iff it contains no units.
If H*x is in H for every x in R, H is a right ideal. If H is a left ideal and a right ideal, H is an ideal.
The intersection of arbitrarily many left ideals is another left ideal. Let H be such an intersection and note that x*H is in every ideal in the set, hence x*H is in the intersection, and x*H is in H. Make the same statement about the sum of two elements in H, and H is a left ideal.
Given a set S of ring elements, the "smallest" left ideal containing S is well defined. Take the intersection of all the left ideals that contain S.
The sum H1+H2, consisting of elements x+y for x in H1 and y in H2, is another left ideal. It includes both H1 and H2, as in H1+0 and 0+H2.
The product of left ideals H1*H2 is the smallest left ideal containing all elements xy from H1 cross H2. That's the definition, but the product operator is usually applied to two sided ideals.
We can characterize the product of two ideals as all finite sums of xy, where x is from the first ideal and y is from the second. This is indeed the smallest ideal containing the pairs xy. Note that the product is contained in the intersection.
Multiply three ideals together and the result is all finite sums of xyz, where variables are taken from their respective ideals. This characterization shows ideal multiplication is associative. It is not usually commutative, unless the base ring is commutative.