= Algebraic structure
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An algebraic structure is a <set> equipped with specified <algebraic operations> satisfying selected axioms. An operation's domain is part of the structure: a <binary operation> on $S$ takes two elements of $S$ and produces another element of $S$. <Groups> specify one such operation, <rings> specify addition and multiplication, and <vector spaces> specify vector addition and scalar multiplication by a separate <field>.
Different choices of operations can give the same underlying <set> different structures. The <integers> form a <group> under addition, since every integer has an additive inverse. They do not form a <group> under multiplication: for example, $2$ has no multiplicative inverse among the <integers>. Even the nonzero <integers> fail that inverse requirement. In contrast, nonzero <rational numbers> form a <group> under multiplication.
Finite multiplication tables make such distinctions visible. Arithmetic modulo $5$ gives a <field>; arithmetic modulo $6$ gives a <commutative ring> with <zero divisors>. In the former, multiplying by any nonzero element permutes all residue classes. In the latter, multiplying by $2$ reaches only $0,2,4$, and $2\cdot3=0$ even though both factors are nonzero.
\Image[/algebra-modular-multiplication-tables.png]
{title=Multiplication tables distinguish a prime field from a ring with zero divisors}
The goal is to extract reusable properties rather than memorize isolated tables. A <homomorphism> preserves the specified operations, and an <isomorphism> shows that two structures have identical operation patterns after relabeling.
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