2 Fixed typo

Following the idea of Andrew of looking at interesting rings to find interesting modules, I suggest you take a look at rings which do not satisfy the IBN (invariant basis number property). Those are rings for which exist left (right) modules $M$ such that $M^n \cong M^m$ for some different positive integers $m,n$; this is, rings for which a notion of "dimension" (rank) of modules cannot be properly defined.

In the late 1950s, W.G. Leavitt ((1),(2),(3)) constructed explicit examples of rings $R$ that do not have the IBN property, that in rigor asks for any two bases (i.e., linearly independent spanning sets) of a free left $R$-module to have the same number of elements, generalizing a well-known property of fields. Noetherian rings and commutative rings are included among the many classes of rings having this property (so that if you really want to find modules quite different in nature to vectorial vector spaces, you must move to the non-commutative setting). It can be shown that if $R$ is a unital ring, then $R^1 \cong R^n$ (as left $R$-modules, for example) for some $n>1$ if and only if there is a set of $2n$ elements in $R$ which produce the appropriate isomorphisms as matrix multiplications by an $n$-row vector and an $n$-column vector with entries in $R$, that is, if there exist elements $x_1, ... , x_n \in R$ and $y_1,..., y_n \in R$ such that $x_iy_j = \delta_{ij}1_R$ for all $i,j$, and $\sum_{i=1}^n y_ix_i = 1_R$. If a unital ring $R$ does not have IBN, it is said that $R$ has module type $(m,n)$, where $m\in {\mathbb N}$ is the minimal number such that $R^m \cong R^n$ for some $n>m$ and $n$ is minimal given $m$. In his seminal paper (3), Leavitt proved that for each pair of positive integers $n>m$ and any field $K$ there always exists a $K$-algebra of module type $(m,n)$, namely the quotient of the unital free associative $K$-algebra in the appropriate number of variables satisfying the relations described above. This algebra is denoted by $L_K(m,n)$ and called the Leavitt $K$-algebra of type $(m,n)$.The $K$-algebras of module type $(m,n)$ need not be unique: for example, if $V$ is an infinite dimensional vector space and $R={\rm End}_K V$ is its ring of endomorphisms, then it is easily seen that $R^1 \cong R^2$; but it can be shown that $R$ is not isomorphic to $L_K(1,2)$.

(1) W.G. Leavitt. Modules over rings of words. Proc. Amer. Math. Soc. 7 (1956), 188-193.

(2) W.G. Leavitt. Modules without invariant basis number. Proc. Amer. Math. Soc. 8 (1957), 322-328.

(3) W.G. Leavitt. The module type of a ring. Trans. Amer. Math. Soc. 42 (1962), 113-130.

1

Following the idea of Andrew of looking at interesting rings to find interesting modules, I suggest you take a look at rings which do not satisfy the IBN (invariant basis number property). Those are rings for which exist left (right) modules $M$ such that $M^n \cong M^m$ for some different positive integers $m,n$; this is, rings for which a notion of "dimension" (rank) of modules cannot be properly defined.

In the late 1950s, W.G. Leavitt ((1),(2),(3)) constructed explicit examples of rings $R$ that do not have the IBN property, that in rigor asks for any two bases (i.e., linearly independent spanning sets) of a free left $R$-module to have the same number of elements, generalizing a well-known property of fields. Noetherian rings and commutative rings are included among the many classes of rings having this property (so that if you really want to find modules quite different in nature to vectorial spaces, you must move to the non-commutative setting). It can be shown that if $R$ is a unital ring, then $R^1 \cong R^n$ (as left $R$-modules, for example) for some $n>1$ if and only if there is a set of $2n$ elements in $R$ which produce the appropriate isomorphisms as matrix multiplications by an $n$-row vector and an $n$-column vector with entries in $R$, that is, if there exist elements $x_1, ... , x_n \in R$ and $y_1,..., y_n \in R$ such that $x_iy_j = \delta_{ij}1_R$ for all $i,j$, and $\sum_{i=1}^n y_ix_i = 1_R$. If a unital ring $R$ does not have IBN, it is said that $R$ has module type $(m,n)$, where $m\in {\mathbb N}$ is the minimal number such that $R^m \cong R^n$ for some $n>m$ and $n$ is minimal given $m$. In his seminal paper (3), Leavitt proved that for each pair of positive integers $n>m$ and any field $K$ there always exists a $K$-algebra of module type $(m,n)$, namely the quotient of the unital free associative $K$-algebra in the appropriate number of variables satisfying the relations described above. This algebra is denoted by $L_K(m,n)$ and called the Leavitt $K$-algebra of type $(m,n)$.The $K$-algebras of module type $(m,n)$ need not be unique: for example, if $V$ is an infinite dimensional vector space and $R={\rm End}_K V$ is its ring of endomorphisms, then it is easily seen that $R^1 \cong R^2$; but it can be shown that $R$ is not isomorphic to $L_K(1,2)$.

(1) W.G. Leavitt. Modules over rings of words. Proc. Amer. Math. Soc. 7 (1956), 188-193.

(2) W.G. Leavitt. Modules without invariant basis number. Proc. Amer. Math. Soc. 8 (1957), 322-328.

(3) W.G. Leavitt. The module type of a ring. Trans. Amer. Math. Soc. 42 (1962), 113-130.