Operations on Ideals
We introduce some notation:
Ideal Generation
Let $R$ be a ring and let $X \subset R$. Then recall $\langle X \rangle$ denotes the ideal generated by $X$, that is, $\cap\{J  X\subset J \triangleleft R\}$. Note the empty set generates the zero ideal.
Set $\mathcal{L}(R) = \{JJ\triangleleft R\}$. This is a poset with respect to $\subset$. Moreover, $\mathcal{L}(A)$ is a complete lattice: for any $S \subset \mathcal{L}(A)$, we have
Sums of Sets
If $X, Y \subset R$ then define $X + Y=\{x+yx \in X,y\in Y\}$. Define $X + y = X + \{y\}$. This is consistent with our notation for cosets. Also note if $J,K\triangleleft R$ then $J+K=\langle J\cup K\rangle$.
More generally, if $\{J_i  i\in I\}$ is a family of ideals in $R$, then define
where only finitely many $x_i$ are nonzero. Then we have
Thus we see finding least upper bounds in $\mathcal{L}(R)$ is equivalent to taking sums of families of ideals.
Products of Ideals and Sets
Now suppose $J \triangleleft R, X\subset A$. Define
We have $J X = \langle a x  a\in J, x \in X\rangle \triangleleft R$.
Define $J x = J \{x\}$. Then we have $ J x = \{a x  a \in J\}$. Note $R x = \langle x \rangle$, the principal ideal generated by $x$.
If $J,K,L \triangleleft R$, then $(J K)L = J (K L)$. More generally, if $J_1,...,J_k \triangleleft R$ then
Powers of Ideals
Powers of $J\triangleleft R$ are defined as follows:
Hence $J^k = \{0\}$ if and only if all products of $k$ elements of $J$ are zero.
Note if $J,K\triangleleft R$ then $J K \subset J\cap K$.
Example:

Let $R=\mathbb{Z}$, $J =\langle m \rangle, K=\langle n\rangle$ where $m,n \in \mathbb{Z}$. Then we have $J\cap K = \langle lcm(m,n)\rangle, J+K=\langle gcd(m,n)\rangle$. Hence the lattice of ideals of $\mathbb{Z}$ can be identified with the lattice
We also have $J K = \langle m n \rangle$, thus $J K = J\cap K$ if and only if $m, n$ are coprime. 2. Let $A =F[x_1,...,x_n]$ for a field $F$. Let $J=\langle x_1,...,x_n\rangle$, so $J$ consists of the polynomials with zero constant term. Then for $m\ge 1$, $J^m$ is precisely the set of the polynomials where each term has degree at least $m$.
The following are easy to verify:

$J(K+L)=J K + J L$

$J\cap(K+L)\supset (J\cap K)+(J\cap L)$

$J\supset K \implies J\cap(K+L) = (J\cap K) +(J\cap L)$ (modular law)

$(J+K)(J\cap K) \subset J K$
We have $(J+K)(J\cap K) \subset J K \subset J\cap K \subset J,K \subset J+K$
(TODO: Hasse diagram)
Example:

In $\mathbb{Z}$, since $lcm(a,gcd(b,c)) = gcd(lcm(a,b),lcm(a,c))$ for nonnegative integers $a,b,c$ we have $J\cap (K+L) = (J\cap K)+(J\cap L)$. Also, as $a b = gcd(a,b)lcm(a,b)$ we have $(J+K)(J\cap K) = J K$.
We say ideals $J,K$ of a ring $R$ are coprime or maximal if $J+K =R$. Equivalently, for some $x\in J, y\in K$ we have $x + y = 1$. Note that if $J, K$ are coprime then $J \cap K = J K$, since we have $J \cap K = (J + K)(J \cap K) \subset J K \subset J \cap K$.