Appendix

Pseudomodules

1. Adjunction of a unit element to a pseudo-ring

Let A be a pseudo-ring (I, § 8, no. 1). On the set $A' = \mathbf{Z} \times A$ we define the following laws of composition:

It is immediately verified that $A'$ with these two laws of composition is a ring in which the element $(1, 0)$ is the unit element. The set $\{0\} \times A$ is a two-sided ideal of $A'$ and $i : x \mapsto (0, x)$ is an isomorphism of the pseudo-ring A onto the sub-pseudo-ring $\{0\} \times A$ by means of which A and $\{0\} \times A$ are identified. $A'$ is called the ring derived from the pseudo-ring A by adjoining a unit element.

If A already has an identity element $\varepsilon$, the element $e = (0, \varepsilon)$ of $A'$ is an idempotent belonging to the centre of $A'$ and such that

Then $(eA', (1 - e)A')$ is a direct decomposition (I, § 8, no. 11) of $A'$ and the ring $(1 - e)A'$ is isomorphic to $\mathbf{Z}$.

2. Pseudomodules

Given a pseudo-ring A with or without a unit element, a left pseudomodule over A is a commutative group E (written additively) admitting A as set of operators and satisfying axioms $(\mathrm{M}_I), (\mathrm{M}_{II})$ and $(\mathrm{M}_{III})$ of § 1, no. 1, Definition 1. Right pseudomodules over A are defined similarly.

Let $A'$ be the ring obtained by adjoining a unit element to A. If E is a left pseudomodule over A, a left $A'$-module structure on E is associated with it by writing, for all $x \in E$ and every element $(n, a) \in A'$,

Axioms $(\mathrm{M}_I)$ to $(\mathrm{M}_{IV})$ of § 1, no. 1, Definition 1 are immediately verified;

moreover, by restricting the set of operators of this module structure to $\{0\} \times A$ (identified with $A$), we obtain on $E$ the pseudomodule structure given initially.

For a subset $M$ of $E$ to be a subgroup with operators of the pseudomodule $E$ (in which case the induced structure is obviously also a left pseudomodule structure over $A$), it is necessary and sufficient that $M$ be a submodule of the associated $A'$-module $E$ and this sub-$A'$-module is associated with the pseudomodule $M$. Moreover, the quotient $A'$-module $E/M$ is then associated with the quotient group with operators $E/M$, which is obviously a pseudomodule over $A$.

If $E, F$ are two pseudomodules over $A$, the homomorphisms $E \to F$ of groups with operators are identical with the $A'$-linear mappings $E \to F$ of the $A'$-modules associated respectively with the pseudomodules $E$ and $F$. If $(E_i)_{i \in I}$ is a family of pseudomodules over $A$, the groups with operators $\prod_{i \in I} E_i$ and $\bigoplus_{i \in I} E_i$ are pseudomodules over $A$ and the associated $A'$-modules are respectively the product and direct sum of the associated $A'$-modules $E_i$. There are analogous results for inverse and direct limits of pseudomodules. The theory of pseudomodules over $A$ can thus be entirely reduced to that of $A'$-modules.

Exercises

See the exercises for Appendix 0.