A ring $A$ is said to be a normed ring if $A$ possesses a norm $\| \cdot \|$ , that is, a non-negative real-valued function $\|\cdot \|:A\to \mathbb{R}$ such that for any $a,b\in A$ ,

1. $\|a\|=0$ iff $a=0$ ,
2. $\|a+b\|\le \|a\|+\|b\|$ ,
3. $\|-a\|=\|a\|$ , and
4. $\|ab\|\le \|a\|\|b\|$ .

Remarks.

• If $A$ contains the multiplicative identity $1$ , then $0<\|1\| \le\|1\|\|1\|$ and so $1\le \|1\|$ .
• However, it is usually required that in a normed ring, $\|1\|=1$ .
• $\|\cdot\|$ defines a metric $d$ on $A$ given by $d(a,b)=\|a-b\|$ , so that $A$ with $d$ is a metric space and one can set up a topology on $A$ by defining its subbasis a collection of $B(a,r):=\lbrace x\in A\mid d(a,x)< r\rbrace$ called open balls for any $a\in A$ and $r>0$ . With this definition, it is easy to see that $\|\cdot\|$ is continuous.
• Given a sequence $\lbrace a_n\rbrace$ of elements in $A$ , we say that $a$ is a limit point of $\lbrace a_n\rbrace$ , if $$\lim_{n\to\infty}\|a_n-a\|=0.$$ By the triangle inequality, $a$ , if it exists, is unique, and so we also write $$a=\lim_{n\to\infty}a_n.$$
• In addition, the last condition ensures that the ring multiplication is continuous.

An algebra $A$ over a field $k$ is said to be a normed algebra if

1. $A$ is a normed ring with norm $\|\cdot\|$ ,
2. $k$ is equipped with a valuation $| \cdot |$ , and
3. $\|\alpha a\|=|\alpha|\|a\|$ for any $\alpha \in k$ and $a\in A$ .

Remarks.

• Alternatively, a normed algebra $A$ can be defined as a normed vector space with a multiplication defined on $A$ such that multiplication is continuous with respect to the norm $\|\cdot\|$ .
• Typically, $k$ is either the reals $\mathbb{R}$ or the complex numbers $\mathbb{C}$ , and $A$ is called a real normed algebra or a complex normed algebra correspondingly.
• A normed algebra that is complete with respect to the norm is called Banach algebra (the underlying field must be complete and algebraically closed), paralleling with the analogy with a Banach space versus a normed vector space.
• Normed rings and normed algebras are special cases of the more general notions of a topological ring and a topological algebra, the latter of which is defined as a topological ring over a field such that the scalar multiplication is continuous.

Bibliography

1

M. A. Naimark: Normed Rings, Noordhoff, (1959).

2

C. E. Rickart: General Theory of Banach Algebras, Van Nostrand, 1960.