Left and Right Inverses of Product
Jump to navigation
Jump to search
Theorem
Let $\struct {S, \circ}$ be a monoid whose identity is $e_S$.
Let $x, y \in S$.
Let:
- $(1): \quad x \circ y$ have a left inverse for $\circ$
- $(2): \quad y \circ x$ have a right inverse for $\circ$.
Then both $x$ and $y$ are invertible for $\circ$.
Proof
Let $z_L$ be the left inverse of $x \circ y$ and $z_R$ be the right inverse of $y \circ x$. Then:
\(\ds z_L \circ \paren {x \circ y}\) | \(=\) | \(\ds e_S\) | by hypothesis | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \paren {z_L \circ x} \circ y\) | \(=\) | \(\ds e_S\) | Associativity of $\circ$ |
\(\ds \paren {y \circ x} \circ z_R\) | \(=\) | \(\ds e_S\) | by hypothesis | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds y \circ \paren {x \circ z_R}\) | \(=\) | \(\ds e_S\) | Associativity of $\circ$ |
Thus $y$ has both a left inverse $z_L \circ x$ and a right inverse $x \circ z_R$.
From Left Inverse and Right Inverse is Inverse:
- $z_L \circ x = x \circ z_R$
and $y$ has an inverse, that is, is invertible.
$\Box$
From the above, we have:
\(\ds \paren {z_L \circ x} \circ y\) | \(=\) | \(\ds e_S\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \paren {x \circ z_R} \circ y\) | \(=\) | \(\ds e_S\) | Left Inverse and Right Inverse is Inverse | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds x \circ \paren {z_R \circ y}\) | \(=\) | \(\ds e_S\) | Associativity of $\circ$ |
and:
\(\ds y \circ \paren {x \circ z_R}\) | \(=\) | \(\ds e_S\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds y \circ \paren {z_L \circ x}\) | \(=\) | \(\ds e_S\) | Left Inverse and Right Inverse is Inverse | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \paren {y \circ z_L} \circ x\) | \(=\) | \(\ds e_S\) | Associativity of $\circ$ |
Thus $x$ has both a left inverse $y \circ z_L$ and a right inverse $z_R \circ y$.
From Left Inverse and Right Inverse is Inverse:
- $y \circ z_L = z_R \circ y$
and $x$ has an inverse, that is, is invertible.
$\blacksquare$
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 4$: Neutral Elements and Inverses: Exercise $4.9 \ \text{(b)}$