## The singular submodule

Posted: December 7, 2011 in Elementary Algebra; Problems & Solutions, Rings and Modules
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Throughout $R$ is a ring with 1 and all modules are left $R$-modules. In Definition 2 in this post, we defined $Z(M),$ the singular submodule of a module $M.$

Problem 1. Let $M$ be an $R$-module and suppose that $N_1, \cdots, N_k$ are submodules of $M.$ Prove that $\bigcap_{i=1}^k N_i \subseteq_e M$ if and only if $N_i \subseteq_e M$ for all $i.$

Solution. We only need to solve the problem for $k = 2.$ If $N_1 \cap N_2 \subseteq_e M,$ then $N_1 \subseteq_e M$ and $N_2 \subseteq_e M$ because both $N_1$ and $N_2$ contain $N_1 \cap N_2.$ Conversely, let $P$ be a nonzero submodule of $M.$ Then $N_1 \cap P \neq \{0\}$ because $N_1 \subseteq_e M$ and therefore $(N_1 \cap N_2) \cap P = N_2 \cap (N_1 \cap P) \neq \{0\}$ because $N_2 \subseteq_e M. \ \Box$

Problem 2. Prove that if $M$ is an $R$-module, then $Z(M)$ is a submodule of $M$ and $Z(R)$ is a proper two-sided ideal of $R.$ In particular, if $R$ is a simple ring, then $Z(R)=\{0\}.$

Solution. First note that $0 \in Z(M)$ because $\text{ann}(0)=R \subseteq_e R.$ Now suppose that $x_1,x_2 \in Z(M).$ Then $\text{ann}(x_1+x_2) \supseteq \text{ann}(x_1) \cap \text{ann}(x_2) \subseteq_e M,$ by Problem 1. Therefore $\text{ann}(x_1+x_2) \subseteq_e M$ and hence $x_1+x_2 \in Z(M).$ Now let $r \in R$ and $x \in Z(M).$ We need to show that $rx \in Z(M).$ Let $J$ be a nonzero left ideal of $R.$ Then $Jr$ is also a left ideal of $R.$ If $Jr = \{0\},$ then $J \subseteq \text{ann}(rx)$ and thus $\text{ann}(rx) \cap J = J \neq \{0 \}.$ If $Jr \neq \{0\},$ then $\text{ann}(x) \cap Jr \neq \{0\}$ because $x \in Z(M).$ So there exists $s \in J$ such that $sr \neq 0$ and $srx = 0.$ Hence $0 \neq s \in \text{ann}(rx) \cap J.$ So $rx \in Z(M)$ and thus $Z(M)$ is a submodule of $M.$ Now, considering $R$ as a left $R$-module, $Z(R)$ is a left ideal of $R,$ by what we have just proved. To see why $Z(M)$ is a right ideal, let $r \in R$ and $x \in Z(R).$ Then $\text{ann}(xr) \supseteq \text{ann}(x) \subseteq_e R$ and so $\text{ann}(xr) \subseteq_e R,$ i.e. $xr \in Z(R).$ Finally, $Z(R)$ is proper because $\text{ann}(1)=\{0\}$ and so $1 \notin Z(R). \ \Box$

Problem 3. Prove that if $M_i, \ i \in I,$ are $R$-modules, then $Z(\bigoplus_{i \in I} M_i) = \bigoplus_{i \in I} Z(M_i).$ Conclude that if $R$ is a semisimple ring, then $Z(R)=\{0\}.$

Solution. The first part is a trivial result of Problem 1 and this fact that if $x = x_1 + \cdots + x_n,$ where the sum is direct, then $\text{ann}(x) = \bigcap_{i=1}^n \text{ann}(x_i).$ The second now follows trivially from the first part, Problem 2 and the Wedderburn-Artin theorem. $\Box$

Problem 4. Suppose that $R$ is commutative and let $N(R)$ be the nilradical of $R.$ Prove that

1) $N(R) \subseteq Z(R);$

2) it is possible to have $N(R) \neq Z(R);$

3) if $Z(R) \neq \{0\},$ then $N(R) \subseteq_e Z(R),$ as $R$-modules or $Z(R)$-modules.

Solution. 1) Let $a \in N(R).$ Then $a^n = 0$ for some integer $n \geq 1.$ Now suppose that $0 \neq r \in R.$ Then $ra^n=0.$ Let $m \geq 1$ be the smallest integer such that $ra^m = 0.$ Then $0 \neq ra^{m-1} \in \text{ann}(a) \cap Rr$ and hence $a \in Z(R).$

2) Let $R_i = \mathbb{Z}/2^i \mathbb{Z}, \ i \geq 1$ and put $R=\prod_{i=1}^{\infty}R_i.$ For every $i,$ let $a_i = 2 + 2^i \mathbb{Z}$ and consider $a = (a_1,a_2, \cdots ) \in R.$ It is easy to see that $a \in Z(R) \setminus N(R).$

3) Let $a \in Z(R) \setminus N(R).$ Then $\text{ann}(a) \cap Ra \neq \{0\}$ and thus there exists $r \in R$ such that $ra \neq 0$ and $ra^2=0.$ Hence $(ra)^2 = 0$ and so $ra \in N(R).$ Thus $0 \neq ra \in N(R) \cap Ra$ implying that $N(R)$ is an essential $R$-submodule of $Z(R).$ Now, we view $Z(R)$ as a ring and we want to prove that $N(R)$ as an essential ideal of $Z(R).$ Again,  let $a \in Z(R) \setminus N(R).$ Then $\text{ann}(a) \cap Ra^2 \neq \{0\}$ and thus there exists $r \in R$ such that $ra^2 \neq 0$ and $ra^3 = 0.$ Let $s = ra \in Z(R).$ Then $(sa)^2=0$ and thus $0 \neq sa \in N(R) \cap Z(R)a$ implying that $N(R)$ is an essential ideal of $Z(R). \ \Box$