Is the assignment operator really "just" an operator?

From the doc: https://ruby-doc.org/docs/ruby-doc-bundle/Manual/man-1.4/syntax.html#assign

Assignment expression are used to assign objects to the variables or such. Assignments sometimes work as declarations for local variables or class constants. The left hand side of the assignment expressions can be either:

variables variables `=' expression

On the right there is an expression, so the result of the expression is assigned to the variable.

So, you should look for expressions (*) before following the precedence.

1 && a = 3 are basically two "chained" expressions:

3 and 1 && 3

Maybe it is more readable as: 1 && a = 3 + 4 where the expressions are 3 + 4 and 1 && 7, see:

1 && a = 3 + 4 #=> 7
1 && 7 #=> 7
res = 1 && a = 3 + 4
res #=> 7

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(*) The precedence table also helps to find the expression (Find the precedence table in the linked doc at the Operator expressions paragraph):

What's above the = in the table "forms" an expression to be assigned by =, what's below does not.

For example:

1 + 3 and 2 + 4 #=> 4
a = 1 + 3 and b = 2 + 4 #=> 4
(a = 1 + 3) and (b = 2 + 4) #=> 4
a = (1 + 3 and b = 2 + 4) #=> 6

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You can also check these examples respect to the precedence table:

1 && 3 #=> 3
1 && a = 3 #=> 3
a #=> 3

3 and 1 #=> 3
3 and b = 1 #=> 3
b #=> 1

2 ** c = 2 + 1 #=> 8
c #=> 3

d = 2 ** 3
d #=> 8

e = 3
e **= 2
e #=> 9