If R is a commutative ring (not necessarily a domain), define t M={m ∈M: r m=0 for some nonzero

step 1 For chapter 2, section 6, question 32, to show that I is not in the range of T, so there would e

If R is a commutative ring (not necessarily a domain),...

If $R$ is a commutative ring (not necessarily a domain), define $t M=\{m \in M: r m=0$ for some nonzero $r \in R\} .$ (i) Let $R=\mathbb{I}_{6}$, and regard $R$ as a module over itself. Prove that $[1] \notin t \mathbb{I}_{6}$. (ii) Prove that $t I_{6}$ is not a submodule of $I_{6}$. Hint. Both $[2],[3] \in t \mathbb{I}_{6}$, but $[3]-[2] \notin t \mathbb{I}_{6}$.

If $R$ is a commutative ring (not necessarily a domain), define $t M=\{m \in M: r m=0$ for some nonzero $r \in R\} .$
(i) Let $R=\mathbb{I}_{6}$, and regard $R$ as a module over itself. Prove that $[1] \notin t \mathbb{I}_{6}$.
(ii) Prove that $t I_{6}$ is not a submodule of $I_{6}$.
Hint. Both $[2],[3] \in t \mathbb{I}_{6}$, but $[3]-[2] \notin t \mathbb{I}_{6}$.

Key Concepts

Commutative Ring

A commutative ring is an algebraic structure in which addition and multiplication are defined and satisfy familiar properties such as associativity, distributivity, and the existence of additive identity and inverses, with the additional requirement that multiplication is commutative. This concept underpins much of algebra, as structures like modules are built over commutative rings, although the ring in question need not be an integral domain, meaning that zero divisors may exist.

Module

A module over a ring is a generalization of the notion of a vector space, where instead of a field the scalars come from an arbitrary ring. Modules allow for the study of linear algebraic properties in a broader context. When the ring is not a domain, the behavior of modules can become more intricate, especially with respect to properties like torsion.

Torsion Element

A torsion element in a module is one that is annihilated by some nonzero element of the ring; that is, there exists a nonzero ring element that, when multiplied by the module element, gives zero. The collection of all torsion elements, often denoted by t(M), forms the 'torsion submodule' in modules over integral domains, but in rings with zero divisors this collection may fail to be a submodule.

Submodule

A submodule is a subset of a module that is closed under the operations of module addition and scalar multiplication by any element of the ring. Submodules play a role analogous to subspaces in vector spaces, but the criteria for being a submodule can be more sensitive in the presence of zero divisors, which can lead to situations where a naturally defined collection, like the torsion elements, does not form a submodule.

Zero Divisor

A zero divisor in a ring is a nonzero element that can multiply with another nonzero element to yield zero. Zero divisors are a key point of study in rings that are not domains, as they affect the behavior of multiplications in modules, particularly in the concept and properties of torsion elements, where the existence of such elements can disrupt standard module properties like closure under subtraction.

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