Simple arithmetic produces some quite
useful results when you can add and multiply values, all of one kind, producing
values of the same kind as the sums and products. I now move on to look at when
we've got some values that can be added to one another, producing values of
the same kind
, and a ringlet (or ring, or even field) by which we can
multiply those values, to scale
them, in one sense or another, to get
values of the original kind.
The easy example of this, given a ringlet on a collection R of values, would be some collection of (partial, if you like) lists with entries in R; the usual pointwise addition for these adds corresponding entries (treating any gap in either as if it has a zero in it; in particular, extending shorter lists with zero entries when adding them to longer lists), so that [1,2,3] + [4,5] = [1+4,2+5,3]; and multiplying by a member of R just scales each entry, so [1,2,3].r = [r, 2.r, 3.r]; the empty list serves as additive identity (or the list whose entries are all zero, when R has a zero, if you don't like partial lists and insist they all have the same length). Note that, to make the addition cancellable, if the ringlet has a zero, comparison must only care about non-zero entries. One can, likewise, use any collection C at all and consider {mappings (R::C)} under the same pointwise addition and scaling; the case of lists justs uses {naturals} or some particular natural as C.
Note that repeated addition of
the multiplicative identity, in any ringlet, generates an image of the positive
naturals {1, 2, 3, …} in the ringlet's centre (so I could safely use
these as entries in my lists, above) – albeit there may be some natural n
for which the ringlet's image of n is an additive identity, i.e. zero, in which
case this image cycles, identifying naturals that differ by multiples of n; I'll
be particularly interested in the case where this doesn't happen, where
the ringlet is said to
have characteristic zero
. This
means that the values by which we can scale include the usual counting numbers
(the naturals); and, when scaling by each
positive natural produces all of our addable values as outputs, this can
naturally be extended to include scaling by the
rationals.
A ringlet homomorphism f from a
ringlet to the automorphism ringlet
of an addition, on some collection M of
values, can be read as a multiplicative action of the ringlet on M. As a
ringlet homomorphism, f necessarily maps the ringlet's multiplicative identity,
1, to the identity f(1) = M on M; and, for any m in M and values r, s of the
ringlet, we get f(r.s, m) = f(r)(f(s, m)) and f(r +s, m) = (f(r) +f(s))(m). As
it maps into the automorphism ringlet, which uses pointwise addition, this last
is f(r +s, m) = f(r, m) +f(s, m); and, as its outputs are automorphisms, with n
also in M, we get f(r, n +m) = f(r, n) +f(r, m). If we interpret f as
a multiplication
of ringlet values on addition values, these look like
(r.s).m = r.(s.m), r.(n+m) = (r.n) +(r.m) and (r +s).m = (r.m) +(s.m), i.e. the
ringlet's multiplicative action plays sensibly with the ringlet's own
multiplication and distributes over both the addition and the ringlet's
addition. The first of these lets us write (r.s).m = r.(s.m) as r.s.m without
introducing any ambituity.
Can we read this multiplicative action (manifestly from the left) also as a right-multiplication ? (We shall want to do this once we get to discussing tensor products, or applying this to an addition on automorphisms, so mapping our ringlet into the automorphisms of those automorphisms.) If we do so, we read (r.s).m = r.(s.m) as r.(m.s) as (m.s).r = m.(s.r) which would be what we read (s.r).m as; so we conflate (r.s).m with (s.r).m, for general values m of the addition and r, s of the ringlet. This only works well if our ringlet is commutative (i.e. if its multiplication is; its addition always is). It could, of course, also work if our homomorphism identified s.r with r.s for all values r, s of our ringlet; reducing the ringlet modulo (: f |) would then give us a ringlet that is commutative, so we may as well work with that.
When such a homomorphism, from a (multiplicatively) commutative ringlet to
the automorphisms of an addition, is interpreted as a multiplicative action of
the ringlet on the addition's values, the resulting structure is described as
a module
over the ringlet; or, when the ringlet is R, as
an R-module
. It is usual to refer to the collection of values of the
addition as the R-module, taking the arithmetic as given.
I shall refer to the values of the ringlet as scalar
values, or just scalars
, and describe their multiplicative action on the
module as scaling
. Thus I can use {scalars} for the collection of values
of the ringlet. In contexts where the ringlet in use is implicit, the
R-qualification (mostly R-
prefixes) on terminology is usually
omitted.
I refer to a module of a division
ringlet R as an R-linear space (or a linear space over R);
when such a ringlet is additively complete (which, as we'll see below, makes its
modules also additively complete, hence additive groups) it's a field and its
modules are orthodoxly called vector spaces (over R, or R
vector spaces
).
Naturally, we reuse the name 0 for each module's additive identity, just as we do for each addition, including each ringlet, that has one; likewise, where negation works we use unary − to denote it.
When our commutative ringlet R has an additive identity, 0, each non-empty R-module must also have an identity, since: for m in the module, r in R, r.m = (r +0).m = r.m +0.m; thus for each n in the module, r.m +n = r.m +0.m +n and cancellation in the module gives n = 0.m +n, whence 0.m must be an additive identity in the module.
If R has an additive inverse for any scalar r (hence also an additive identity) we get, for any m in M, 0 = 0.m = (−r +r).m = (−r).m +r.m whence (−r).m is an additive inverse for r.m, for each m in M, and we can write (−r).m = −(r.m), making it unambiguous to refer to −r.m without the parentheses. When the multiplicative identity in R has an additive inverse, −1, this gives us an additive inverse for each m in M, making M additively complete.
When R is a ring (so additively complete), its modules have to be additively complete, since R does indeed have an additive identity and an additive inverse for its multiplicative identity.
If a module M has an additive identity, 0, consider any m in M and scalar r; we have r.m = r.(m +0) = r.m +r.0; for any n in M, we thus have r.m +n = r.m +r.0 +n and cancelling in M gives n = r.0 +n; this being true for all n in (commutative) M, we infer that r.0 is an additive identity in M, hence r.0 = 0. The additive identity in the module is unchanged by scaling, limiting the extent to which cancellation of scaling could possibly work.
If the module M has an additive inverse for some m in M then, for any scalar r, we have 0 = r.(−m +m) = r.(−m) +r.m, making r.(−m) an additive inverse for r.m; we can thus write r.(−m) = −(r.m), albeit we do need the parentheses unless r has an additive inverse, to make −(r.m) = (−r).m (as discussed above). In particular, negation commutes with scaling; we shall see (below) how this lets us tacitly complete our ringlet's addition if it isn't already complete.
Given a relation (N: f :M) between R-modules N, M I'll say that
f respects R-scaling
precisely if r is scalar and f relates n to m
implies f relates r.n to r.m
. In particular, this says that every
R-multiple of any left or right value is likewise a left or right value; and,
when a module M subsumes some collection S, we can construe S as a relation (M:
:M); then S respects R-scaling precisely if it subsumes {r.s: r is scalar, s is
in S}. (This last necessarily subsumes S, as r = 1 shows, so the two are equal
if S subsumes it.) Such a collection is also described as closed under
R-scaling, since R-scaling any member of it gives a member of it.
Given two R-modules M and N: a mapping (N: f |M) that respects addition (i.e. is a homomorphism of the additions on M and N) and R-scaling (i.e. f(r.m) = r.f(m) for every scalar r and every m in M) is called an R-module homomorphism or simply R-homomorphism; when f is also a monic (N| :M), hence iso, it is called a module isomorphism. When R is descalable, so M and N are linear spaces, R-homomorphisms are described as R-linear or simply linear maps.
Given two R-modules M and N, H = {R-module homomorphisms (N: |M)} forms an
R-module under the usual pointwise addition and scaling of relations. When N is
R's addition, construed as an R-module, H is referred to as
the dual
of M, written dual(M).
Scaling by any scalar is trivially a module autormorphism, since scalar multiplication is commutative. The above specification implies that a module automorphism is precisely an automorphism of the addition that commutes with every scaling.
When the module has an additive identity, necessarily unchanged by scaling, its zero mapping ({0}: |M) is trivially (and boringly) an automorphism. Since every automorphism of an addition maps the additive identity to itself, every automorphism of our module (including every scaling) commutes with this zero automorphism.
When the module's addition is complete, its negation mapping (: −m ←m :) is an automorphism of the addition, as usual. We saw above that it also commutes with scaling, r.(−m) = −(r.m) for any scalar r; thus it is a module automorphism also. Furthermore, for any automorphism f, we have 0 = f(0) = f(−m +m) = f(−m) +f(m) implying f(−m) = −f(m), so negation commutes with all automorphisms.
Even when our ringlet is additively incomplete, if the module itself is additively complete we can form a commuting sub-ringlet of its automorphisms comprising the scalings, the zero map and the composite of each scaling with negation; these form a ring (that is, an additively complete ringlet) and we can interpret our module as a module over this ring, instead of over the original ringlet. Since our chosen automorphisms commute with all the same automorphisms that the original scalings do, we get exactly the same automorphisms for our module when considered as a module over the ring as when considered as a module over the original ringlet.
Notice that we started by embedding a ringlet in an addition's automorphisms; the image of that ringlet is our ringlet of scalings; we then select, from among the addition's automorphisms, the ones that commute with these scalings; these are the automorphisms of the module. These necessarily form a sub-ringlet of the addition's automorphism ringlet; this needn't (and typically won't) be a commutative sub-ringlet; however, within it we can select its centre, the collection of automorphisms that commute with all (module) automorphisms. This is a sub-ringlet of the module automorrphisms and has the scalings as a sub-ringlet; we can naturally extend our interpretation of the module to understand it as a module over the centre of the automorphism ringlet we got from our original scalings; chosing to interpret the centre as our scalings doesn't change what module we're studying, as it still has the same automorphisms as before.
We can likewise take any commuting sub-ringlet of the automorphisms of our addition and use it as the ringlet over which to read our addition as a module; we thereby select the automorphisms of the addition that commute with those in this sub-ringlet as the automorphisms of the module. Choice of which sub-ringlet to use shall affect which automorphisms of the addition qualify as automorphisms of the module. One example of making such a choice leads to obtaining a complexified ring from a ringlet.
In particular, because an additively complete module over a ringlet can always be handled in terms of the centre of the ring of module automorphisms over tha ringlet, which will additively complete our ringlet for us, we lose nothing in our understanding of the module by the ringlet being additively incomplete; thus, if the ringlet we started from is adequately described by some sub-ringlet of it that gains some advantages by being additively incomplete (e.g. the simplifications that arise from using the positive sub-ringlet in an ordered ring), it suffices to consider the module over that sub-ringlet, reaping the benefits of additive completion from the module while keeping the benefits that the sub-ringlet provided.
Every ringlet contains an image of the naturals; in the ringlet of automorphisms of an addition, these are repeated addition; the addition's characteristic is zero precisely if this image of the naturals is injective (i.e. it doesn't map any positive natural to the addition's zero automorphism). So, we can construe any character zero addition as a module over the positive naturals (which have some nice properties) and let the addition's own authormorphisms complete that to
This being true for any addition that has the relevant automorphisms, any time we have a module over the integers, positive ratios or rationals, we can equally deal with it strictly as a module over the naturals, without introducing ratios or integers, by using the induced automorphisms of our module in their place; these would serve as integers and rationals without needing to explicitly construct collections of specific entities, divorced from any particular module, to represent them.
I must look into whether we can do similar with complexification, using polynomial completion among the automorphisms to give us the complexified extension of a ringlet. Likewise, presumably, one can extend to the reals if the automorphisms include all their Cauchy completions, regardless of whether the ringlet our module is given to be over has the same.
When the ringlet has a conjugation,
c, a relation (N: f :M) between its modules can interact with scaling via that
instead of simply respecting scaling: I'll say
f conjugates
scaling precisely if f relates n to m
implies, for every value r of the ringlet, f relates c(r).n to r.m
. As c
is self-inverse this incidentally also implies that f relates r.n to c(r).m. An
automorphism of the additions on M and N is
an conjugate-homomorphism
between M and N as R-modules
precisely if it conjugates R-scaling. When R is a division ringlet, its
conjugate-homomorphisms are described as conjugate-linear
maps. Many authors refer to these as antilinear
, a term I've surely also
used outside the more orderly part of my writings.
A composite of (arbitrarily many) homomorphisms mixed with an even number of conjugate-homomorphisms between modules is a homomorphism; composing module homomorphisms mixed with with an odd number of conjugate-homomorphisms gives an conjugate-homomorphism. The collection of conjugate-homomorphisms between any two given modules forms a module over the same ringlet.
Consider an conjugate-homomorphism (: f |M) whose outputs are homomorphisms
(N: |M); thus f takes two values of m and combines them to make a value of N.
If we scale f's two inputs, the first's scaling gets conjugated, the seconds is
not; so, for scalars r, s and values u, v of M, we have f(r.u, s.v) =
c(r).s.f(u, v). When r = s is non-zero, this scales f(u, v) by the positive
scalar c(r).r; when N is {scalars} this makes it possible for each u in M to
have f(u, u) real within {scalars} – in contrast to a homomorphism
producing homomorphisms, or an conjugate-homomorphism producing
conjugate-homomorphisms, where any f(u, u) that's real would give f(s.u, s.u)
scaled by s.s or c(s.s), which won't generally be real. So
conjugate-homomorphisms that produce homomorphisms provide a way to collapse
down
the complexity of a ringlet to its real part. For that to work, we
need f(u +v, u +v) = f(u, u) + f(v, v) + f(u, v) + f(v, u) to be real; when f(u,
u) and f(v, v) are, this needs f(v, u) and f(u, v) to add up to a real; at least
for the complex case, this is assured when f(v, u) and f(u, v) are mutually
conjugate. An conjugate-homomorphism (dual(M): f |M) is described
as a hermitian form
precisely if f(u, v) = c(f(v, u)) for
each u, v.
When a hermitian form f has f(v, v) positive (not just real) for
all non-zero v, the form is described
as positive-definite
; likewise, if all such f(v, v) are
negative as negative-definite
. If no f(v, v) is negative (but some f(v,
v) for non-zero v may be zero) it is
positive semi-definite
; likewise if none are positive, it
is negative semi-definite.
Given an R-module M (an introduction that tacitly introduces R as a commutative ringlet and introduces an addition and R-scaling on M), if M subsumes N which is closed within M under M's addition (whose restriction to N is necessarily cancellable) and N respects R-scaling, then N is described as a sub-module of M (or a sub-R-module, on the rare occasions when discussing a module over more than one ringlet). When R is descalable, so M is a linear space (or even a vector space, if R is a field), each sub-module of M is known as a sub-space.
Given an R-module M and a sub-module N of it, we can define an equivalence on M that relates m to p precisely if there are n, q in N for which m+n = p+q. Since scaling all parties by a common scalar r gives r.m +r.n = r.p +r.q with r.n and r.q in N, the equivalence relates r.m to r.p whenever it relates m to p; so the equivalence respects R-scaling. If, as well as m equivalent to p, we have w equivalent to b, so m+n = p+q and w+u = b+d for some n, p, u, d in N, then we get (m+w)+(n+u) = (p+b)+(q+d) with each of n+u and q+d in N (because it was closed under addition) hence m+w equivalent to p+q; so this equivalence also respects our addition on M. When M is considered modulo this equivalence it is written as M/N; because the equivalence respects addition and R-scaling, M/N is also an R-module, known as the quotient of M by N or, when M is taken as given, as N's quotient (in M).
The kernel and image of a module homomorphism mapping are defined in the usual way for homomorphisms between additions and are, as usual, closed under addition. If we scale a member f(m) of Image(f) by any scalar r, we get r.f(m) = f(r.m) also in Image(f). If we scale m in Ker(f) by scalar r we have f(r.m) = r.f(m) with f(m) an additive identity in N, hence so is r.f(m) by the reasoning above; hence r.m is also in Ker(f). Thus Image(f) and Ker(f) are closed under addition and R-scaling, making Image(f) a sub-module of N and Ker(f) a sub-module of M. When N has no additive identity (hence neither has M, since its image under f would need to be one), the kernel is necessarily empty; otherwise, it is ({0}&on;f: m←m :). The image is simply (|f:M) or indeed (|f:), which is a collection because if it relates u to n then there is some m in M for which u = f(m) = n and f is a mapping, so u = n.
Notice that a commutative ringlet's addition is always a module of the ringlet, using the ringlet's multiplication as scaling. Indeed, in the same way, any ideal (including the ringlet itself) of any ringlet is a module over any commutative sub-ringlet. Likewise, the ringlet of polynomials over a given ringlet R is a module over any commutative sub-ringlet of R.
For any collection I and any R-module M, we can make an R-module of the usual addition on {mappings (M: :I)} via pointwise multiplication, r.f = (: r.f(i) ←i :I), in the usual way; and likewise for {mappings (M: |I)}. In particular, we can apply this with I = {naturals} or with some specific natural as I. Thus (optionally partial) lists or sequences whose entries fall in some module of a ringlet (including the ringlet's own addition) form a module under any commutative sub-ringlet.
Given an R-automorphism (M: f |M), we can make M (tacitly an R-module) a module over polynomials in R by making p.m = p(f, m) for each polynomial p over R and m in M. In such a case, the sub-modules over polynomials are precisely the R-submodules of M that f maps into themselves, i.e. the R-submodules W of V for which (W: f |W). When R is descalable (or indeed a field), making M a linear space (or vector space), this makes M a module over R's ringlet (or ring) of polynomials.
We can treat any cancellable combnator, interpreted as addition on its collection G of values, as a module over {positive naturals} using the multiplicative action these induce via repeated addition; every homomorphism from it, as an addition, is then then a module homomorphism; and every sub-addition a sub-module. We can naturally extend this to a module over {naturals} if G has an additive identity, via 0.g being that identity in G; and to a module over {integers} if G's addition is complete – hence a group: it has an identity and inverses. When G is a group, its submodules over {integers} are precisely its subgroups; and module homomorphisms over {integers} in this sense are precisely the usual group homomorphisms, in so far as we interpret the other end of the homomorphism as a module over {integers} in the matching way.
In any R-module M, if a positive natural n has scale(n) = (M| n.m ←m |M), onto all of M, we can interpret its reverse as an inverse scaling; when it is a mapping (i.e. scale(n) is monic), we can read it as an automorphism that commutes with all automorphisms, allowing us to extend R to include 1/n as an honourary member; when this works for each positive natural n, we can extend our ringlet to include all positive ratios. (If the module is additively complete and has characteristic zero, we thus get to extend our ringlet to include the ring of rationals.)
When a ringlet has a conjugation and resulting norm, we can use this to induce a topology and augment our natural and rational scalings with any limit points that may be present in the ringlet, so as to obtain as much of the reals as it contains. Even where the ringlet lacks limit points, if the module has all limit points for some distance measure (e.g. a positive-definite hermitian form), then we can find scalings of it that fill in the ringlet's missing limit points, thereby making it a module over a real-completed version of the ringlet we started with.
Further study required: I suppose that there is some algebraic completeness property that will enable us to extend the ringlet to the algebraic numbers within the centre of a module's ringlet of automorphisms; and, when this is combined with the topological completion just alluded to (that produces reals), we shall get the complex numbers.
These various ways of infering useful structure in the centre of a module's ringlet of automorphisms, from properties of the module, free us from the need to require these properties of the ringlet, over which we understand it as a module. Some properties of the ringlet do constrain which automorphisms of the addition shall be understood as module automorphisms, which in turn constrains the centre of the ringlet of automorphisms; so we do need the ringlet to have these properties; but additive completeness is not one of them, since the module's own addition will supply additive completeness, when it is relevant, regardless of whether our chosen ringlet has it. Since additive completeness obliges us to split many pieces of analysis into cases distinguished by sign (whether a value is positive, negative or zero), which complicates and obscures analysis, it is at times more of an impediment than a help. For this reason, I opt to work with {positives} as ringlet wherever I can, while exploiting the larger ringlet of scalings, which is the centre of the module's ringlet of automorphisms, wherever I have need of it.
Written by Eddy.