The factor group Q/Z and the circle group are also injective Z-modules. The factor group Z/nZ for n > 1 is injective as a Z/nZ-module, but not injective as an abelian group.
What is module M?
A module M is Noetherian if L(M) satisfies ACC or, equivalently, if L (M) satisfies the maximum condition. From: Ring Theory, 83, 1991.
What is a divisible module?
A module over a unit ring is called divisible if, for all which are not zero divisors, every element of can be “divided” by , in the sense that there is an element in such that . This condition can be reformulated by saying that the multiplication by defines a surjective map from to .
Is Q za projective Z module?
Because Z is a PID, Q is also a free Z-module But It’s not. Because for all submodules of Q \ {0}, they are not linearly independent over Z. And thus the only independent submodule of Q is {0}, which cannot span the whole Q. So Q cannot find a basis of Z-module.
What does Injective mean in math?
one-to-one function
In mathematics, an injective function (also known as injection, or one-to-one function) is a function f that maps distinct elements to distinct elements; that is, f(x1) = f(x2) implies x1 = x2. In other words, every element of the function’s codomain is the image of at most one element of its domain.
What are Z modules?
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).
What are the modules?
A module is a separate unit of software or hardware. Typical characteristics of modular components include portability, which allows them to be used in a variety of systems, and interoperability, which allows them to function with the components of other systems. The term was first used in architecture.
What does it mean for a group to be divisible?
From Wikipedia, the free encyclopedia. In mathematics, especially in the field of group theory, a divisible group is an abelian group in which every element can, in some sense, be divided by positive integers, or more accurately, every element is an nth multiple for each positive integer n.
What does it mean for a group to be torsion free?
a group in which every element other than the identity has infinite order. …
Is Q free as AZ module?
But Q is not a cyclic Z module (it is divisible, so it is not isomorphic to Z, the only infinite cyclic Z-module. So Q cannot be free.
Are projective modules flat?
Every projective module is flat. The converse is in general not true: the abelian group Q is a Z-module which is flat, but not projective. Conversely, a finitely related flat module is projective.
What is injective function example?
Injective function or injection of a function is also known as one one function and is defined as a function in which each element has one and only one image. This every element is associated with atmost one element. f:N→N:f(x)=2x is an injective function, as.
What is an injective module in Algebra?
Injective module. In mathematics, especially in the area of abstract algebra known as module theory, an injective module is a module Q that shares certain desirable properties with the Z-module Q of all rational numbers.
What is the difference between indecomposable injective and injective modules?
Every injective module is uniquely a direct sum of indecomposable injective modules, and the indecomposable injective modules are uniquely identified as the injective hulls of the quotients R / P where P varies over the prime spectrum of the ring.
Do all submodules have to be injective?
In general, submodules, factor modules, or infinite direct sums of injective modules need not be injective.
What is the injective hull of the Z -module?
Two examples are the injective hull of the Z -module Z / pZ (the Prüfer group ), and the injective hull of the k [ x ]-module k (the ring of inverse polynomials). The latter is easily described as k [ x, x−1 ]/ xk [ x ]. This module has a basis consisting of “inverse monomials”, that is x−n for n = 0, 1, 2, ….
