Biserial projectives in the beta-two socle quotient #
The canonical quotient by all non-simple projective-injective socles has right-middle arity at most two at every vertex. The finite radical recursion therefore makes every indecomposable projective of that quotient biserial, with separated uniserial radical branches.
For an ideal-annihilated module with simple top on both sides of the quotient equivalence, biseriality over the quotient algebra is equivalent to biseriality over the ambient algebra.
Under the same simple-top hypotheses, the stronger decomposition into two uniserial branches with zero intersection is also invariant under the ideal-quotient equivalence.
Every indecomposable projective of the canonical simultaneous socle quotient has radical equal to the internal direct sum of at most two uniserial branches.
Every indecomposable projective of the canonical simultaneous socle quotient is biserial.
Every ambient indecomposable projective not belonging to the deleted non-simple projective-injective family is biserial. Such a projective is an ordinary surviving quotient projective, so its quotient-algebra biseriality transports back through the annihilated full subcategory.
A selected non-simple projective-injective right ideal is biserial. Its socle quotient is one of the replacement projectives in the simultaneous quotient. The separated radical branches of that replacement transport back to the literal socle quotient and then lift across its simple essential socle.
If the right AR middle-term bound is at most two, every primitive right ideal in the chosen presentation is a biserial module.
The right beta bound at most two makes the chosen primitive-projective presentation biserial on both sides. The left ideals are the right ideals of the opposite presentation, whose beta bound is Gabriel's left/right comparison.
For an algebra carrying a complete duplicate-free primitive-projective presentation, the beta-two bound produces a literal special-biserial bound-quiver presentation.
The beta-two bound implies special-biseriality whenever the ambient algebra itself is supplied with the complete primitive presentation.
For an arbitrary representation-finite algebra, the canonical basic endomorphism algebra carries a literal special-biserial presentation under the beta-two bound.
The final Morita transport, isolated from the representation-theoretic argument. Once the canonical basic endomorphism algebra is connected to the ambient algebra by Mathlib's all-module Morita witness, its literal special-biserial presentation gives the manuscript's Morita-invariant predicate for the ambient algebra.
For an arbitrary representation-finite algebra, the beta-two bound implies the Morita-invariant special-biserial predicate. The all-module Morita witness comes from the basic projective generator of the contragredient skeleton; its target is the opposite algebra whose literal special-biserial presentation is supplied by the opposite primitive-projective presentation.