Construction For An Arbitrary Family of Modules
One should notice a clear similarity between the definitions of the direct sum of two vector spaces and of two abelian groups. In fact, each is a special case of the construction of the direct sum of two modules. Additionally, by modifying the definition one can accommodate the direct sum of an infinite family of modules. The precise definition is as follows (Bourbaki 1989, §II.1.6).
Let R be a ring, and {Mi : i ∈ I} a family of left R-modules indexed by the set I. The direct sum of {Mi} is then defined to be the set of all sequences where and for cofinitely many indices i. (The direct product is analogous but the indices do not need to cofinitely vanish.)
It can also be defined as functions α from I to the disjoint union of the modules Mi such that α(i) ∈ Mi for all i ∈ I and α(i) = 0 for cofinitely many indices i. These functions can equivalently be regarded as finitely supported sections of the fiber bundle over the index set I, with the fiber over being .
This set inherits the module structure via component-wise addition and scalar multiplication. Explicitly, two such sequences (or functions) α and β can be added by writing for all i (note that this is again zero for all but finitely many indices), and such a function can be multiplied with an element r from R by defining for all i. In this way, the direct sum becomes a left R-module, and it is denoted
It is customary to write the sequence as a sum . Sometimes a primed summation is used to indicate that cofinitely many of the terms are zero.
Read more about this topic: Direct Sum Of Modules
Famous quotes containing the words construction, arbitrary and/or family:
“No real “vital” character in fiction is altogether a conscious construction of the author. On the contrary, it may be a sort of parasitic growth upon the author’s personality, developing by internal necessity as much as by external addition.”
—T.S. (Thomas Stearns)
“If you should rise from Nowhere up to Somewhere,
From being No one up to being Someone,
Be sure to keep repeating to yourself
You owe it to an arbitrary god....”
—Robert Frost (1874–1963)
“What we often take to be family values—the work ethic, honesty, clean living, marital fidelity, and individual responsibility—are in fact social, religious, or cultural values. To be sure, these values are transmitted by parents to their children and are familial in that sense. They do not, however, originate within the family. It is the value of close relationships with other family members, and the importance of these bonds relative to other needs.”
—David Elkind (20th century)