The Ideal Vacuum

Jessica K. Sklar, Shawna Holman, and Matthew Fishburn

In this project, we present a collection of photographic metaphors for abstract algebraic concepts. The series has a two-fold purpose: first, to be available as a learning aid for abstract algebra students, and second, to be an attractive work of art that can perhaps engage the general public in conversations about higher-level mathematics. Click on an image to read about the concept it represents.

You are encouraged to contribute your own photographic metaphors or suggestions of metaphors. Please email any suggestions to sklarjk@plu.edu.

Essential

Essential: Let R be a ring. A submodule N of an R-module M is said to be essential in M if whenever L≠0 is a submodule of M, then N∩L≠0.

The idea behind this photograph is to portray the "expansiveness" of an essential submodule. To use a linguistic metaphor, an essential submodule N of a module M is a submodule that has its fingers in the pots of every one of M's submodules: if L is any nontrivial submodule of M, then N and L must have a nontrivial intersection. N need not contain many elements, however; its significance in M is due to its "density" in M, so to speak, rather than to its size. This photograph attempts to convey this density: the tree (our submodule) fills the picture (our module), leaving no room in which other submodules can play without its leaves getting in the way..

(See dual photograph, Superfluous.)

Superfluous

Superfluous: Let R be a ring. A submodule N of an R-module M is said to be superfluous in M if whenever L ≠ M is a submodule of M, then N+L ≠ M.

In contrast to the tree in the photograph Essential, this tree is hardly space-filling. As in the case of essentialness, superfluity has more to do with density than with size: a submodule is superfluous if it is, in some sense, sparse in the module. Specifically, a submodule is superfluous in M if it is so "small" that it cannot generate all of M even when taken together with any other proper R-submodule. Using trees in the photographs for the concepts of both "essential" and "superfluous" emphasizes these adjectives' natures as mathematical duals.

(See dual photograph, Essential.)

Artinian

Artinian: Let R be a ring. An R-module M is said to be artinian if whenever M₁, M₂, . . . is a sequence of R-submodules of M with

M₁ ⊇ M₂ ⊇ . . .

then there exists a positive integer n such that M_m = M_n for every m ≥ n.

Each doll in this sequence represents one of the M_i. While this is perhaps a fairly straightforward visual metaphor for an artinian module, it is important to note that while this photograph implies that the M_i are decreasing in "size" (cardinality), this need not be the case: for instance, all of the submodules could be countably infinite. Rather, we have that each M_i contains M_i+1. While these dolls obviously do not contain one another in the photograph, the use of matryoshka dolls gives a nod to the notion of containment. As we move from left to right, the dolls eventually become the same in size, reflecting the fact that a descending sequence of submodules of an artinian module must eventually become stationary: that is, after some point in the sequence, the M_i are all identical.

(See dual photograph, Noetherian.)

Noetherian

Noetherian: Let R be a ring. An R-module M is said to be noetherian if whenever M₁, M₂, . . . is a sequence of R-submodules of M with

M₁ ⊆ M₂ ⊆ . . .

then there exists a positive integer n such that M_m = M_n for every m ≥ n.

Noetherian is the mathematical dual of artinian. In a noetherian module, each submodule M_i is contained in (rather than contained by) the submodule M_i+1. Eventually the sequence of submodules becomes stationary. In an analogous situation to that of the photograph Artinian, this is represented by the matryoshka dolls eventually (as we move from left to right) becoming constant in size.

(See dual photograph, Artinian.)

Invertible Function

Invertible function: A function f from a set X to a set Y is said to be invertible if there exists a function g from Y to X such that f(g(y)) = y and g(f(x)) = x for every y in Y and x in X.

In this image, we see a standard hourglass; applying our function f corresponds to flipping the hourglass. When f is applied, the grains of sand (elements) in the top of the hourglass (set X) are sent to the bottom of the hourglass (set Y). This action can then be reversed: we can send the grains of sand (now elements of Y) back to set X by, literally, inverting the hourglass—this action corresponds to applying function g, the inverse of f. While flipping an hourglass is not exactly a function (it does not, in fact, send grains of sand bijectively to other grains of sand), it captures the notion of invertibility: it is an action that can be reversed, ad infinitum, to send elements back from whence they came.

Socle

Socle: Let R be a ring. The socle of an R-module M is the (unique) maximal semisimple R-submodule of M.

A plant frog plays the role in a vase that a socle plays in an artinian module. Every submodule of an artinian module contains a simple submodule; thus, the socle of the module forms, in a sense, the module's "base" (hence, the name "socle," an architectural term referring to a plinth used to support a pedestal or column). A plant frog pierces (is contained in) flowers (submodules), and forms the "base" of a flower arrangement (module).

Ideal

Ideal: Let R be a ring. A subset I of R is said to be an ideal of R if it's a subgroup of R, and has the property that for every r in R, we have

rI ⊆ I and Ir ⊆ I.

One way to recall the definition of an ideal is to remember that ideals "suck." If I is an ideal of ring R, then when any element of R is multiplied by any element of I, the resulting element must be in I; in a sense, the ideal uses multiplication to draw elements into itself. Granted, like our metaphor for an Invertible Function, this isn't a perfect explanation of the situation: if an element r of R is multiplied by an element of I, r itself does not become a member of I. However, this colorful description of I's role in R can be extremely pedagogically useful: people remember the concept, because they love saying that things "suck."

Hence, the photograph.