In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a ring. The concept of module generalizes also the notion of abelian group, since the abelian groups are exactly the modules over the ring of integers.

Like a vector space, a module is an additive abelian group, and scalar multiplication is distributive over the operation of addition between elements of the ring or module and is compatible with the ring multiplication.

Modules are very closely related to the representation theory of groups. They are also one of the central notions of commutative algebra and homological algebra, and are used widely in algebraic geometry and algebraic topology.

Introduction and definition[edit]

Motivation[edit]

In a vector space, the set of scalars is a field and acts on the vectors by scalar multiplication, subject to certain axioms such as the distributive law. In a module, the scalars need only be a ring, so the module concept represents a significant generalization. In commutative algebra, both ideals and quotient rings are modules, so that many arguments about ideals or quotient rings can be combined into a single argument about modules. In non-commutative algebra, the distinction between left ideals, ideals, and modules becomes more pronounced, though some ring-theoretic conditions can be expressed either about left ideals or left modules.

Much of the theory of modules consists of extending as many of the desirable properties of vector spaces as possible to the realm of modules over a "well-behaved" ring, such as a principal ideal domain. However, modules can be quite a bit more complicated than vector spaces; for instance, not all modules have a basis, and even for those that do (free modules) the number of elements in a basis need not be the same for all bases (that is to say that they may not have a unique rank) if the underlying ring does not satisfy the invariant basis number condition, unlike vector spaces, which always have a (possibly infinite) basis whose cardinality is then unique. (These last two assertions require the axiom of choice in general, but not in the case of finite-dimensional spaces, or certain well-behaved infinite-dimensional spaces such as L^p spaces.)

Formal definition[edit]

Suppose that R is a ring, and 1 is its multiplicative identity. A left R-module M consists of an abelian group (M, +) and an operation · : R × M → M such that for all r, s in R and x, y in M, we have

$r\cdot (x+y)=r\cdot x+r\cdot y$
$(r+s)\cdot x=r\cdot x+s\cdot x$
$(rs)\cdot x=r\cdot (s\cdot x)$
$1\cdot x=x.$

The operation · is called scalar multiplication. Often the symbol · is omitted, but in this article we use it and reserve juxtaposition for multiplication in R. One may write _RM to emphasize that M is a left R-module. A right R-module M_R is defined similarly in terms of an operation · : M × R → M.

Authors who do not require rings to be unital omit condition 4 in the definition above; they would call the structures defined above "unital left R-modules". In this article, consistent with the glossary of ring theory, all rings and modules are assumed to be unital.^[1]

An (R,S)-bimodule is an abelian group together with both a left scalar multiplication · by elements of R and a right scalar multiplication ∗ by elements of S, making it simultaneously a left R-module and a right S-module, satisfying the additional condition (r · x) ∗ s = r ⋅ (x ∗ s) for all r in R, x in M, and s in S.

If R is commutative, then left R-modules are the same as right R-modules and are simply called R-modules.

Examples[edit]

If K is a field, then K-vector spaces (vector spaces over K) and K-modules are identical.
If K is a field, and K[x] a univariate polynomial ring, then a K[x]-module M is a K-module with an additional action of x on M that commutes with the action of K on M. In other words, a K[x]-module is a K-vector space M combined with a linear map from M to M. Applying the structure theorem for finitely generated modules over a principal ideal domain to this example shows the existence of the rational and Jordan canonical forms.
The concept of a Z-module agrees with the notion of an abelian group. That is, every abelian group is a module over the ring of integers Z in a unique way. For n > 0, let n ⋅ x = x + x + ... + x (n summands), 0 ⋅ x = 0, and (−n) ⋅ x = −(n ⋅ x). Such a module need not have a basis—groups containing torsion elements do not. (For example, in the group of integers modulo 3, one cannot find even one element which satisfies the definition of a linearly independent set since when an integer such as 3 or 6 multiplies an element, the result is 0. However, if a finite field is considered as a module over the same finite field taken as a ring, it is a vector space and does have a basis.)
The decimal fractions (including negative ones) form a module over the integers. Only singletons are linearly independent sets, but there is no singleton that can serve as a basis, so the module has no basis and no rank.
If R is any ring and n a natural number, then the cartesian product Rⁿ is both a left and right R-module over R if we use the component-wise operations. Hence when n = 1, R is an R-module, where the scalar multiplication is just ring multiplication. The case n = 0 yields the trivial R-module {0} consisting only of its identity element. Modules of this type are called free and if R has invariant basis number (e.g. any commutative ring or field) the number n is then the rank of the free module.
If M_n(R) is the ring of n × n matrices over a ring R, M is an M_n(R)-module, and e_i is the n × n matrix with 1 in the (i, i)-entry (and zeros elsewhere), then e_iM is an R-module, since re_im = e_irm ∈ e_iM. So M breaks up as the direct sum of R-modules, M = e₁M ⊕ ... ⊕ e_nM. Conversely, given an R-module M₀, then M₀^⊕n is an M_n(R)-module. In fact, the category of R-modules and the category of M_n(R)-modules are equivalent. The special case is that the module M is just R as a module over itself, then Rⁿ is an M_n(R)-module.
If S is a nonempty set, M is a left R-module, and M^S is the collection of all functions f : S → M, then with addition and scalar multiplication in M^Sdefined pointwise by (f + g)(s) = f(s) + g(s) and (rf)(s) = rf(s), M^S is a left R-module. The right R-module case is analogous. In particular, if R is commutative then the collection of R-module homomorphisms h : M → N (see below) is an R-module (and in fact a submodule of N^M).
If X is a smooth manifold, then the smooth functions from X to the real numbers form a ring C^∞(X). The set of all smooth vector fields defined on Xform a module over C^∞(X), and so do the tensor fields and the differential forms on X. More generally, the sections of any vector bundle form a projective module over C^∞(X), and by Swan's theorem, every projective module is isomorphic to the module of sections of some bundle; the categoryof C^∞(X)-modules and the category of vector bundles over X are equivalent.
If R is any ring and I is any left ideal in R, then I is a left R-module, and analogously right ideals in R are right R-modules.
If R is a ring, we can define the opposite ring R^op which has the same underlying set and the same addition operation, but the opposite multiplication: if ab = c in R, then ba = c in R^op. Any left R-module M can then be seen to be a right module over R^op, and any right module over R can be considered a left module over R^op.
Modules over a Lie algebra are (associative algebra) modules over its universal enveloping algebra.
If R and S are rings with a ring homomorphism φ : R → S, then every S-module M is an R-module by defining rm = φ(r)m. In particular, S itself is such an R-module.

Submodules and homomorphisms[edit]

Suppose M is a left R-module and N is a subgroup of M. Then N is a submodule (or more explicitly an R-submodule) if for any n in N and any r in R, the product r ⋅ n (or n ⋅ r for a right R-module) is in N.

If X is any subset of an R-module M, then the submodule spanned by X is defined to be ${\textstyle \langle X\rangle =\,\bigcap _{N\supseteq X}N}$ where N runs over the submodules of Mwhich contain X, or explicitly ${\textstyle \left\{\sum _{i=1}^{k}r_{i}x_{i}\mid r_{i}\in R,x_{i}\in X\right\}}$ , which is important in the definition of tensor products.^[2]

The set of submodules of a given module M, together with the two binary operations + and ∩, forms a lattice which satisfies the modular law: Given submodules U, N₁, N₂ of M such that N₁ ⊂ N₂, then the following two submodules are equal: (N₁ + U) ∩ N₂ = N₁ + (U ∩ N₂).

If M and N are left R-modules, then a map f : M → N is a homomorphism of R-modules if for any m, n in M and r, s in R,

f(r\cdot m+s\cdot n)=r\cdot f(m)+s\cdot f(n)

.

This, like any homomorphism of mathematical objects, is just a mapping which preserves the structure of the objects. Another name for a homomorphism of R-modules is an R-linear map.

A bijective module homomorphism f : M → N is called a module isomorphism, and the two modules M and N are called isomorphic. Two isomorphic modules are identical for all practical purposes, differing solely in the notation for their elements.

The kernel of a module homomorphism f : M → N is the submodule of M consisting of all elements that are sent to zero by f, and the image of f is the submodule of N consisting of values f(m) for all elements m of M.^[3] The isomorphism theorems familiar from groups and vector spaces are also valid for R-modules.

Given a ring R, the set of all left R-modules together with their module homomorphisms forms an abelian category, denoted by R-Mod (see category of modules).

Types of modules[edit]

Finitely generated: An R-module M is finitely generated if there exist finitely many elements x₁, ..., x_n in M such that every element of M is a linear combination of those elements with coefficients from the ring R.
Cyclic: A module is called a cyclic module if it is generated by one element.
Free: A free R-module is a module that has a basis, or equivalently, one that is isomorphic to a direct sum of copies of the ring R. These are the modules that behave very much like vector spaces.
Projective: Projective modules are direct summands of free modules and share many of their desirable properties.
Injective: Injective modules are defined dually to projective modules.
Flat: A module is called flat if taking the tensor product of it with any exact sequence of R-modules preserves exactness.
Torsionless: A module is called torsionless if it embeds into its algebraic dual.
Simple: A simple module S is a module that is not {0} and whose only submodules are {0} and S. Simple modules are sometimes called irreducible.^[4]
Semisimple: A semisimple module is a direct sum (finite or not) of simple modules. Historically these modules are also called completely reducible.
Indecomposable: An indecomposable module is a non-zero module that cannot be written as a direct sum of two non-zero submodules. Every simple module is indecomposable, but there are indecomposable modules which are not simple (e.g. uniform modules).
Faithful: A faithful module M is one where the action of each r ≠ 0 in R on M is nontrivial (i.e. r ⋅ x ≠ 0 for some x in M). Equivalently, the annihilator of M is the zero ideal.
Torsion-free: A torsion-free module is a module over a ring such that 0 is the only element annihilated by a regular element (non zero-divisor) of the ring, equivalently rm = 0 implies r = 0 or m = 0.
Noetherian: A Noetherian module is a module which satisfies the ascending chain condition on submodules, that is, every increasing chain of submodules becomes stationary after finitely many steps. Equivalently, every submodule is finitely generated.
Artinian: An Artinian module is a module which satisfies the descending chain condition on submodules, that is, every decreasing chain of submodules becomes stationary after finitely many steps.
Graded: A graded module is a module with a decomposition as a direct sum M = ⨁_x M_x over a graded ring R = ⨁_x R_x such that R_xM_y ⊂ M_x+y for all x and y.
Uniform: A uniform module is a module in which all pairs of nonzero submodules have nonzero intersection.

Further notions[edit]

Relation to representation theory[edit]

A representation of a group G over a field k is a module over the group ring k[G].

If M is a left R-module, then the action of an element r in R is defined to be the map M → M that sends each x to rx (or xr in the case of a right module), and is necessarily a group endomorphism of the abelian group (M, +). The set of all group endomorphisms of M is denoted End_Z(M) and forms a ring under addition and composition, and sending a ring element r of R to its action actually defines a ring homomorphism from R to End_Z(M).

Such a ring homomorphism R → End_Z(M) is called a representation of R over the abelian group M; an alternative and equivalent way of defining left R-modules is to say that a left R-module is an abelian group M together with a representation of R over it. Such a representation R → End_Z(M) may also be called a ring action of R on M.

A representation is called faithful if and only if the map R → End_Z(M) is injective. In terms of modules, this means that if r is an element of R such that rx = 0 for all x in M, then r = 0. Every abelian group is a faithful module over the integers or over some ring of integers modulo n, Z/nZ.

Generalizations[edit]

A ring R corresponds to a preadditive category R with a single object. With this understanding, a left R-module is just a covariant additive functor from Rto the category Ab of abelian groups, and right R-modules are contravariant additive functors. This suggests that, if C is any preadditive category, a covariant additive functor from C to Ab should be considered a generalized left module over C. These functors form a functor category C-Mod which is the natural generalization of the module category R-Mod.

Modules over commutative rings can be generalized in a different direction: take a ringed space (X, O_X) and consider the sheaves of O_X-modules (see sheaf of modules). These form a category O_X-Mod, and play an important role in modern algebraic geometry. If X has only a single point, then this is a module category in the old sense over the commutative ring O_X(X).

One can also consider modules over a semiring. Modules over rings are abelian groups, but modules over semirings are only commutative monoids. Most applications of modules are still possible. In particular, for any semiring S, the matrices over S form a semiring over which the tuples of elements from S are a module (in this generalized sense only). This allows a further generalization of the concept of vector space incorporating the semirings from theoretical computer science.

Over near-rings, one can consider near-ring modules, a nonabelian generalization of modules.^{[citation needed]}

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"SET" in the KJV Bible

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Deuteronomy 17:15chapter context similar meaning copy save: Thou shalt in any wise set him king over thee, whom the LORD thy God shall choose: one from among thy brethren shalt thou set king over thee: thou mayest not set a stranger over thee, which is not thy brother.
Ezekiel 7:20chapter context similar meaning copy save: As for the beauty of his ornament, he set it in majesty: but they made the images of their abominations and of their detestable things therein: therefore have I set it far from them.
Numbers 10:21chapter context similar meaning copy save: And the Kohathites set forward, bearing the sanctuary: and the other did set up the tabernacle against they came.
Ezekiel 24:3chapter context similar meaning copy save: And utter a parable unto the rebellious house, and say unto them, Thus saith the Lord GOD; Set on a pot, set it on, and also pour water into it:
1 Kings 7:21chapter context similar meaning copy save: And he set up the pillars in the porch of the temple: and he set up the right pillar, and called the name thereof Jachin: and he set up the left pillar, and called the name thereof Boaz.
Jeremiah 51:12chapter context similar meaning copy save: Set up the standard upon the walls of Babylon, make the watch strong, set up the watchmen, prepare the ambushes: for the LORD hath both devised and done that which he spake against the inhabitants of Babylon.
Jeremiah 31:21chapter context similar meaning copy save: Set thee up waymarks, make thee high heaps: set thine heart toward the highway, even the way which thou wentest: turn again, O virgin of Israel, turn again to these thy cities.
Psalms 91:14chapter context similar meaning copy save: Because he hath set his love upon me, therefore will I deliver him: I will set him on high, because he hath known my name.
Hosea 2:3chapter context similar meaning copy save: Lest I strip her naked, and set her as in the day that she was born, and make her as a wilderness, and set her like a dry land, and slay her with thirst.
Ezekiel 4:2chapter context similar meaning copy save: And lay siege against it, and build a fort against it, and cast a mount against it; setthe camp also against it, and set battering rams against it round about.
Hebrews 12:2chapter context similar meaning copy save: Looking unto Jesus the author and finisher of our faith; who for the joy that was setbefore him endured the cross, despising the shame, and is set down at the right hand of the throne of God.
2 Samuel 14:30chapter context similar meaning copy save: Therefore he said unto his servants, See, Joab's field is near mine, and he hath barley there; go and set it on fire. And Absalom's servants set the field on fire.
1 Kings 20:12chapter context similar meaning copy save: And it came to pass, when Benhadad heard this message, as he was drinking, he and the kings in the pavilions, that he said unto his servants, Set yourselves in array. And they set themselves in array against the city.
2 Chronicles 2:18chapter context similar meaning copy save: And he set threescore and ten thousand of them to be bearers of burdens, and fourscore thousand to be hewers in the mountain, and three thousand and six hundred overseers to set the people a work.
1 Samuel 26:24chapter context similar meaning copy save: And, behold, as thy life was much set by this day in mine eyes, so let my life be much set by in the eyes of the LORD, and let him deliver me out of all tribulation.
Nehemiah 4:13chapter context similar meaning copy save: Therefore set I in the lower places behind the wall, and on the higher places, I even set the people after their families with their swords, their spears, and their bows.
Daniel 3:12chapter context similar meaning copy save: There are certain Jews whom thou hast set over the affairs of the province of Babylon, Shadrach, Meshach, and Abednego; these men, O king, have not regarded thee: they serve not thy gods, nor worship the golden image which thou hast set up.
1 Samuel 2:8chapter context similar meaning copy save: He raiseth up the poor out of the dust, and lifteth up the beggar from the dunghill, to set them among princes, and to make them inherit the throne of glory: for the pillars of the earth are the LORD'S, and he hath set the world upon them.
Numbers 2:17chapter context similar meaning copy save: Then the tabernacle of the congregation shall set forward with the camp of the Levites in the midst of the camp: as they encamp, so shall they set forward, every man in his place by their standards.
Zechariah 3:5chapter context similar meaning copy save: And I said, Let them set a fair mitre upon his head. So they set a fair mitre upon his head, and clothed him with garments. And the angel of the LORD stood by.
Ezekiel 15:7chapter context similar meaning copy save: And I will set my face against them; they shall go out from one fire, and another fire shall devour them; and ye shall know that I am the LORD, when I set my face against them.
Ezekiel 4:3chapter context similar meaning copy save: Moreover take thou unto thee an iron pan, and set it for a wall of iron between thee and the city: and set thy face against it, and it shall be besieged, and thou shalt lay siege against it. This shall be a sign to the house of Israel.
Mark 8:6chapter context similar meaning copy save: And he commanded the people to sit down on the ground: and he took the seven loaves, and gave thanks, and brake, and gave to his disciples to set before them; and they did set them before the people.
2 Chronicles 13:3chapter context similar meaning copy save: And Abijah set the battle in array with an army of valiant men of war, even four hundred thousand chosen men: Jeroboam also set the battle in array against him with eight hundred thousand chosen men, being mighty men of valour.
Exodus 40:4chapter context similar meaning copy save: And thou shalt bring in the table, and set in order the things that are to be set in order upon it; and thou shalt bring in the candlestick, and light the lamps thereof.
Ezekiel 23:24chapter context similar meaning copy save: And they shall come against thee with chariots, wagons, and wheels, and with an assembly of people, which shall set against thee buckler and shield and helmet round about: and I will set judgment before them, and they shall judge thee according to their judgments.
Daniel 3:3chapter context similar meaning copy save: Then the princes, the governors, and captains, the judges, the treasurers, the counsellors, the sheriffs, and all the rulers of the provinces, were gathered together unto the dedication of the image that Nebuchadnezzar the king had set up; and they stood before the image that Nebuchadnezzar had set up.
1 Samuel 9:24chapter context similar meaning copy save: And the cook took up the shoulder, and that which was upon it, and set it before Saul. And Samuel said, Behold that which is left! set it before thee, and eat: for unto this time hath it been kept for thee since I said, I have invited the people. So Saul did eat with Samuel that day.
Ezekiel 26:20chapter context similar meaning copy save: When I shall bring thee down with them that descend into the pit, with the people of old time, and shall set thee in the low parts of the earth, in places desolate of old, with them that go down to the pit, that thou be not inhabited; and I shall set glory in the land of the living;
Psalms 2:6chapter context similar meaning copy save: Yet have I set my king upon my holy hill of Zion.

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You searched for

"ELEMENTS" in the KJV Bible

Module (mathematics)

Introduction and definition[edit]

Motivation[edit]

Formal definition[edit]

Examples[edit]

Submodules and homomorphisms[edit]

Types of modules[edit]

Further notions[edit]

Relation to representation theory[edit]

Generalizations[edit]

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"SET" in the KJV Bible

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